Some Integer Sequences  

This page gives terms in some of the integer sequences mentioned on my numbers and large numbers pages. Many of these sequences also have their own pages.

Contents

Categories of Sequences With Their Own Pages
Sequences With Pretty Illustrations
Software Related to Numbers and Sequences
Main Table
Longer Lists of Sequence Terms

My recent article Is This Sequence Interesting?

(with cartoons!)


Categories of Sequences With Their Own Pages

I have also written pages discussing each of the following special topics:

2nd-order linear recurrence sequences, a category that includes the Fibonacci sequence and many others.

Sequences Seen in Digits of Rational Fractions: This page also discusses generating functions and methods of deriving a fraction, a sequence, and/or a generating function given one of the others.

The MCS Sequences are a broader class of recurrence relation; this page lists over 1000 sequences with the simplest recurrence formulas.

Floretions, a special type of multidimensional numeric quantity sharing properties of quaternions and sedenions.

Mutually coprime sequences, most of which also grow very quickly.

The dyadic operators page covers all sequences that are found in the infinite square arrays representing the common integer arithmetic operations (addition, multiplication and exponentiation) and their extension to a 4th operator (called tetration, hyper4, or powerlog).

Accelerating Sequences my efforts to define a sequence that starts out slow and speeds up as it goes along.

Generalized Cullen and Woddall numbers

Sequences Related to the Collatz 3X+1 iteration

The Narayana triangle and several related triangles

Kaprekar sequences

A few special sequences related to the Mandelbrot set (more at Mu-Ency)

Fermat numbers, and Prime Factors of Fermat Numbers

Other new work that I plan to submit to OEIS soon.


Sequences With Pretty Illustrations


My Software Related to Numbers and Sequences

RIES is a super-seeker for floating-point values, that returns results ordered by progressively better match and progressively greater complexity.

mcsfind is a super-seeker for integer sequences that explores the space of MCS definitions, and returns results ordered by progressively greater formula complexity.

Hypercalc, for those times when you want to work with really large quantities (e.g. "How many digits are there in 257885161-1?") but don't need to actually see all the digits. Hypercalc eats power-towers like 10101034 for lunch.

Sloandora is an experimental project to provide auto-recommendation browsing for the OEIS, based on a full substring-concordance metric.


Main Table

Ordering Used in the Table

I use the ordering originally established by Sloane [1].

Here the sequences are presented in lexicographic order, as if minus signs and any leading 0 or 1 terms were ignored. For example, "0, 1, 1, -1, -2, 1, 5, -5, -26, 29, ..." is treated as if it starts "2, 1, 5, 5, 26, 29, ...".

A052154 : Coefficents of Lemniscates for Mandelbrot Set, or Binary Trees of Limited Height

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 1, 0, 0, 1, 1, 2, 5, 0, 0, 0, 1, 1, 2, 5, 6, 0, 0, 0, 1, 1, 2, 5, 14, 6, 0, 0, 0, 1, 1, 2, 5, 14, 26, 4, 0, 0, 0, 1, 1, 2, 5, 14, 42, 44, 1, 0, 0, 0, 1, 1, 2, 5, 14, 42, 100, 69, 0, 0, 0, 0, ...

This sequence has its own page here.

OEIS entry links to the mu-ency title page and to my A052154 page.

A020916 : "Molecules" : A Restricted Class of Permutations

0, 0, 1, 2, 0, 0, 24, 96, 0, 0, 10000, 60736, 0, 0, 20511168, 168661760, 0, 0, 134002359296, 1398597049856, 0, 0, 2146989255011328, 27232259080056832, 0, 0, ...

This sequence has its own page here.

A137560 : Coefficents of Lemniscates for Mandelbrot Set, or Binary Trees of Limited Height

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 5, 6, 6, 4, 1, 0, 1, 1, 2, 5, 14, 26, 44, 69, 94, 114, 116, 94, 60, 28, 8, 1, 0, 1, 1, 2, 5, 14, 42, 100, 221, 470, 958, 1860, 3434, 6036, 10068, 15864, 23461, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, ...

This sequence has its own page here.

OEIS entry links to the mu-ency lemniscates page.

A171882 : Iterated Exponential Function (higher Hyper4)

1,1,0,1,1,1,1,2,1,0,1,3,4,1,1,1,4,27,16,1,0,1,5,256,7625597484987,65536,1,1,1,6,3125, 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096, ...

This function is discussed extensively on my hyper4 page.

A039754 : Classes of binary block-codes of codeword length N and with K{<=}2N valid values

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 6, 3, 3, 1, 1, 1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1, 1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, ...

Parts of this sequence shows up in the triangle discused on my A005646 page.

A171881 : Iterated Exponential Function (lower Hyper4)

0, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 27, 16, 1, 1, 5, 256, 19683, 256, 1, 1, 6, 3125, 4294967296, 7625597484987, 65536, 1, 1, 7, 46656, 298023223876953125, 340282366920938463463374607431768211456, 443426488243037769948249630619149892803, 4294967296, 1, 1, 8, 823543, ...

This function is discussed extensively on my hyper4 page.

A000001 : Number of groups of order N.

1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, ...

The sequence starts at N=1. There is always a Cyclic group with a single operation that maps each element onto exactly one other element, such that you have to repeat that operation N times to get the Identity operation. For N=4 there is the Klein four-group, the product of two order-2 cyclic groups (representable as, e.g., the four binary numbers 00, 01, 10, 11 and the XOR operation). For more examples, see List of small groups.

A101211 : Run-length-encoded Binary

1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, ...

I wrote this in the margin of my HIS[1] in the early 1990's but thought it not worth submitting, Leory Quet submitted it independently many years later as A101211.

A000032 : Lucas Numbers beginning with 2, 1

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, ...

This Fibonacci-like sequence is sometimes begun with 1, 3 (omitting the initial 2).

A019473 : Still-Lifes with N cells in Conway's game of Life

0, 0, 0, 2, 1, 5, 4, 9, 10, 25, 46, 121, 240, 619, 1353, 3286, 7773, 19044, 45759, 112243, 273188, 672172, 1646147, 4051711, ...

This sequence has its own page here.

I am listed as author. I believe I sent this to NJAS in the initial email with sequences to add after his 1995 book.

A000005 : Number of divisors of N

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, ...

A000009 : Partitions of N into distinct parts; or into odd parts.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, ...

A092188 : Smallest positive integer M such that 2^3^4^5^...^NM mod N

2, 2, 4, 2, 2, 1, 8, 8, 2, 2, 8, 5, 8, 2, 16, 2, 8, 18, 12, 8, 2, 16, 8, 2, 18, 26, 8, 11, 2, 2, 32, 2, 2, 22, 8, 31, 18, 5, 32, 2, 8, 27, 24, 17, 16, 8, 32, 43, 2, 2, 44, 45, 26, 2, 8, 56, ...

This sequence, Sloane's A092188, has its own page here.

Links to my page on the topic, and mentions that I extended the sequence in 200404.

A000010 : Euler totient function

1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, 32, 24, 52, 18, 40, 24, 36, 28, 58, 16, 60, 30, 36, 32, 48, 20, 66, 32, 44, ...

Counts integers smaller than and relatively prime to N. Discussed here.

A080611 : Smallest base with divisibiliy tests for first N primes

2, 2, 4, 6, 21, 155, 441, 2925, 10165, 342056, 2781505, 10631544, 163886800, 498936010, 5163068911, ...

My numbers page is in the links.

Circuits on the Order-N Dragon Curve

1, 2, 2, 5, 7, 10, 16, 21, ...

Email me if you want a diagram and explanation (my existing notes are hand-written and filed away, but I can render a new diagram)

A005206 : Hofstadter's G-sequence

0, 1, 1, 2, 3, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, ...

G0 = 0; GN = N - GGN-1 for N>0. Appears in Gödel,_Escher,_Bach.

A005185 : Hofstadter's Q-sequence

1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, 11, 11, 12, 12, 12, 12, 16, 14, 14, 16, 16, 16, 16, 20, 17, 17, 20, 21, 19, 20, 22, 21, 22, 23, 23, 24, 24, 24, 24, 24, 32, 24, 25, 30, 28, ...

Q1 = Q2 = 1; QN = QN-QN-1 + QN-QN-2 for N>2.

A005374 : Hofstadter's H-sequence

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14, 14, 15, 16, 17, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 24, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 32, 33, 33, 34, 35, 35, ...

H0 = 0; HN = N - HHHN-1 for N>0. Appears in Gödel,_Escher,_Bach.

A000027 : Natural numbers

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ...

OEIS entry links to my numbers page (along with several other general number-oriented websites).

A160818 : Equals average of all permutations of its digits

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 777, 888, 999, 1111, 2222, 3333, 4444, 5555, ...

This sequence has its own page here.

A162002 : 2N ≡ 22N mod N

1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 15, 16, 17, 18, 20, 24, 26, 28, 30, 32, 34, 36, 40, 42, 43, 44, 46, 48, 51, 52, 56, 58, 60, 64, 68, 70, 72, 76, 78, 80, 84, 85, 88, 90, 96, 100, 102, 104, 112, 120, ...

This sequence has its own page here.

A006874 : Period-N Continental mu-atoms in Mandelbrot Set

1, 1, 2, 3, 4, 6, 6, 9, 10, 12, 10, 22, 12, 18, 24, 27, 16, 38, 18, 44, 36, 30, 22, 78, 36, 36, 50, 66, 28, 104, 30, 81, 60, 48, 72, 158, 36, 54, 72, 156, 40, 156, 42, 110, 152, 66, 46, 270, 78, 140, 96, 132, 52, 230, 120, 234, 108, 84, 58, 456, 60, 90, 228, 243, 144, 260, ...

This sequence is discussed here and here.

OEIS entry links to the mu-ency title page.

Record setter for divisibility tests (2 way with composites)

2, 3, 4, 6, 9, 12, 16, 21, 25, 36, 60, 81, 85, 120, 225, 240, 336, 360, 361, 481, 540, 720, 721, 841, 1080, 1260, 1261, 1680, 2520, 4200, 5040, 5041, 6721, 7560, 9361, 10080, 10081, 15120, ...

This seqeunce is discussed in the numbers entry for 21.

A006875 : Period-N Non-seed mu-atoms in Mandelbrot Set

0, 1, 2, 3, 4, 7, 6, 12, 12, 23, 10, 51, 12, 75, 50, 144, 16, 324, 18, 561, 156, 1043, 22, 2340, 80, 4119, 540, 8307, 28, 17521, 30, 32928, 2096, 65567, 366, 135432, 36, 262179, ...

This sequence is discussed here and here.

OEIS entry links to the mu-ency title page.

A000040 : Prime numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, ...

A000041 : Partitions

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, ...

Sloane's A000041, another combinatoric sequence like the Bell numbers; "ways to put N indistinguishable balls into one or more indistinguishable urns". More terms here.

A000043 : Mersenne exponents: primes p such that 2p-1 is prime.

2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, ...

See here for a more complete list.

A000045 : Fibonacci numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, ...

More terms here.

A006888 : An = An-1 + An-2 An-3

0, 0, 1, 1, 1, 2, 3, 5, 11, 26, 81, 367, 2473, 32200, 939791, 80570391, 30341840591, 75749670168872, 2444729709746709953, 2298386861814452020993305, 185187471463742319884263934176321, 5618934645754484318302453706799174724040986, 425632451384909715242581951982860838627778260930154571891, 1040556299367291626141472184594831289773562749596746466784696205046709264397, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

I discovered this with the Casio 8500fx calculator, and submitted it (up to the 30341840591 term) to NJAS along with several other sequences, in the early 1990's. Here I include the initial two 0 terms that are not in the OEIS entry. Michel ten Voorde later added 4 terms, and the larger ones are from me.

Henry Bottomley found the asymptotic approximation AN {~=} 1.60119...1.32471795...N where the second constant is the sole real solution to (X^3}=X+1.

Gerald McGarvey found that at N goes to infinity, AN/AN-1AN-5 approaches 1. This is related to the 1.32471795... in the Bottomley formula.

A110389 : Product of all preceding terms, minus one (beginning with 2,3)

2, 3, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608869, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068029, ...

This sequence is discussed here.

A006877 : Successive values of X1 for Collatz 3X+1 iteration that set a new record for Np

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, ...

This sequence is discussed here.

I calculated this sequence on an Apple ][ shortly after the publication of Gödel,_Escher,_Bach and submitted it to N.J.A. Sloane in 1994.

A171878 : An = An-1 + An-2 An-3 + An-4An-5

0, 0, 0, 0, 1, 2, 3, 6, 13, 33, 120, 765, 4831534, 55040353993453427047, 410186270246002225336426103593500672000000000000055040353997149550557, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

A056569 : Row sums of Fibonomial triangle A010048

1, 2, 3, 6, 14, 42, 158, 756, 4594, 35532, ...

This sequence is discussed in the number entry for 158.

A069354 : Lowest base with easy test for divibility by N primes

2, 3, 6, 15, 66, 210, 715, 7315, 38571, 254541, 728365, 11243155, 58524466, 812646121, 5163068911, 58720148851, ...

This sequence, Sloane's A069354, has its own page here.

A006884 : Successive values of X1 for Collatz 3X+1 iteration that set a record for Xmax

1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, 1042431, 1212415, 1441407, 1875711, 1988859, ...

This sequence is discussed here.

I calculated this sequence on an Apple ][ shortly after the publication of Gödel,_Escher,_Bach and submitted it to N.J.A. Sloane in 1994.

A000058 : Product of all preceding terms, plus one

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, 165506647324519964198468195444439180017513152706377497841851388766535868639572406808911988131737645185443, ...

This sequence is discussed here.

A006882 or MCS213030 : Double Factorials

1, 1, 2, 3, 8, 15, 48, 105, 384, 945, 3840, 10395, 46080, 135135, 645120, 2027025, 10321920, 34459425, ...

Combined odd and even double factorials. There are a few types of permutations that are counted by this sequence. Any two adjacent terms, multiplied together, gives an ordinary factorial.

In Sloane's original book the odd and even values were segregated into separate sequences (A000165 and A001147, which are sequences M1878 and M3002 in the book) but this sequence was not present. I always considered this to be the more intutive interpretation of "double facorials", so I submitted the sequence in the early 1990's and am listed as the author.

A030124 : Hofstadter Figure-Figure Sequence "S"

2, 4, 5, 6, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, ...

Appears in Gödel,_Escher,_Bach; its terms are both the first differences and the complement in the positive integers of the terms of A005228.

A171880 : An = An-1 + An-2 An-3 + An-4 An-5An-6

0, 0, 0, 1, 1, 1, 2, 4, 7, 16, 46, 166, 1014, 47066, 12348246366, 66716521529543607970475115226, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

A171874 : An = An-1 + An-2 An-3 + An-4An-5

0, 0, 0, 1, 1, 2, 4, 7, 16, 46, 174, 3311, 268446771, 401906756202069927727330981, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

An = An-1 + An-2 An-3 + An-4 An-5An-6 + An-7 An-8An-9An-10

0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 4, 7, 16, 47, 168, 1033, 47347, 12616687331, 95252198578077627625561945222, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1, and also uses the "lower hyper4" operator defined here.

A001200 : Number of linear geometries on n (unlabeled) points.

1, 1, 1, 2, 3, 5, 10, 24, 69, 384, 5250, 232929, 28872973, ...

One of the "hard" sequences from Sloane's first book[1]. It represents a very difficult problem; as of 1973 the terms were only known as far as the "384". The 28872973 result is from 1999.

A000055 : Unlabeled Unrooted Trees

1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, ...

This is some discussion of this sequence on my A005646 page.

A059929 : Fib(N) × Fib(N+2)

0, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 20737, 54288, 142130, 372099, 974170, 2550408, 6677057, 17480760, 45765226, 119814915, 313679522, ...

MCS12921029 : A0 = 0; A1 = 1; A2 = 0; AN = 2 AN-1 + 2 AN-2 - AN-3

A001013 : Jordan-Polya numbers: one or more factorials multipled together

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, ...

Each of these represents the number of permutations you can get by rearranging one or more groups of objects, where the objects in each group stay together. For example if you have 3 apples and 4 oranges, you get a total of 3! × 4! = 6×24 = 144 permutations. If you also allow rearranging the groups themselves (putting the pranges before the apples or vice-versa) that's like adding another group, in this case a group of size 2, for a total of 2×6×24 = 288 permutations. More terms here.

A000124 : Central polygonal numbers 1+N(N+1)/2

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, ...

Not to be confused with Hogben's Centered Polygonal Numbers (A002061).

A001006 : Motzkin N-paths:

1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, ...

This sequence enumerates paths from (0,0) to (N,0) that do not descend below x=0 and that use only the steps U = (1,1), F = (1,0) and D = (1,-1). It appeared during the early design process of my equation-finder ries. (The standard definition, "The number of ways of drawing any number of nonintersecting chords joining N (labeled) points on a circle", would seem to be another sequence entirely, yet is not.)

A014221 : 2N

0, 1, 2, 4, 16, 65536, ...

The next term after 65536 is 2.00352993...×1019728; see this discussion related to the Ackermann function.

My page on A094358 is listed as a link.

Product of all preceding terms, minus 1

2, 5, 9, 89, 8009, 64152089, 4115490587216009, 16937262773463574696951813104089, ...

This sequence is discussed here. (Not in OEIS)

Product of all preceding terms, plus one

2, 5, 11, 111, 12211, 149096311, 22229709804712410, 494159998001727075769152612720511, ...

This sequence is discussed here. (Not in OEIS)

A002559 : Markoff (or Markov) numbers

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, 1597, 2897, 4181, 5741, 6466, 7561, 9077, 10946, 14701, 28657, 33461, 37666, 43261, 51641, 62210, 75025, 96557, 135137, 195025, 196418, 294685, 426389, 499393, 514229, 646018, 925765, ...

The set of all values of X, Y and Z that appear in solutions to X2 + Y2 + Z2 = 3XYZ.

A000108 : Catalan Numbers

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, ...

The Catalan numbers (Sloane's A000108) Related to the Narayana triangle and the Mandelbrot lemniscates. More terms are given below.

A000110 : Bell numbers

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, ...

Sloane's A000110, this sequence shows up in many combinatoric applications; the simplest description is "ways to put N distinguishable balls into one or more indistinguishable urns". More terms below

A100140 : Largest Denominator of Greedy Egyptian Fraction Sum for M/N

2, 6, 4, 20, 3, 231, 24, 45, 20, 4070, 12, 2145, 231, 120, ...

This sequence, Sloane's A100140, has its own page here.

Links to my A100140 page and lists me as author (I submitted this in 200411, during the initial Filene's investigation of greedy Egyptian fractions)

A045619 : Product of two or more consecutive integers

0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, ...

This sequence is discussed here.

&A002378 : Oblong or pronic numbers: N(N+1) = N2+N

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, ...

MCS1764 : AK = K2 + K (score: 3.3)

A001654 : Golden rectangle numbers (product of two consecutive Fibonacci numbers)

0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895, 12816, 33552, 87841, 229970, 602070, 1576239, 4126648, 10803704, 28284465, 74049690, 193864606, 507544127, ...

MCS100945 : A0 = 0; A1 = 1; A2 = 2; AN = 2 AN-1 + 2 AN-2 - AN-3

A001181 : Baxter Permutations, or Subdivided Rectangle Floorplans Allowing Pinwheels

0, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, ...

Illustrations of the partitioned rectangles for N from 1 to 4 are on this page.

A000142 or MCS30992 : Factorials

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, ...

Note that 0!=1. See also A006882.

A001113 : Digits of e

2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0, 4, 5, 2, 3, 5, 3, 6, 0, 2, 8, 7, 4, 7, 1, 3, 5, 2, 6, 6, 2, 4, 9, 7, 7, 5, 7, 2, 4, 7, 0, 9, ...

A006883 : 1/N has repeating decimal of N-1 digits

2, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, ...

I am listed as author; this is one of the sequences in my original list of things to add to Sloane's 1973 book. Note that it is the same as A001913 except for the initial 2. As a child I thought that 2 "obviously" belonged in this sequence because its decimal representation has a sequence of one digit (0) that repeats.

A054670 : Laurent series for mapping onto boundary of Mandelbrot set (Denominators)

2, 8, 4, 128, 1, 1024, 16, 32768, 1, 262144, 32, 4194304, ...

Lists me as author, and links to my mu-ency page on the Laurent series for the Maple code.

A051254 : Mills primes

2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ...

These primes have a pattern: 23+3=11; 113+30=1361; 13613+6=2521008887; 25210088873+80=16022236204009818131831320183; and so on. They are related to the theorem by Mills stating that there is some constant A such that A3N, rounded down to the nearest integer, is a prime for all positive values of N. This particular sequence is related to the hypothesis that there is a minimum value of A that can be found if we assume that there is a prime between any two consecutive cubes. To date, highest known prime in this sequence is 1.7505...×1020561.

A006885 : Successive record values of Xmax for Collatz 3X+1 iteration

1, 2, 16, 52, 160, 9232, 13120, 39364, 41524, 250504, 1276936, 6810136, 8153620, 27114424, 50143264, 106358020, 121012864, 593279152, 1570824736, 2482111348, 2798323360, 17202377752, ...

This sequence is discussed here.

I calculated this sequence on an Apple ][ shortly after the publication of Gödel,_Escher,_Bach and submitted it to N.J.A. Sloane in 1994.

A000796 : Digits of π (pi)

3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, ...

A005267 : product of all preceding terms, minus one

3, 2, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608869, 11392521832807516835658052968328096177131218666695418950023483907701862019030266123104859068029, ...

This sequence is discussed here.

A000203 : Sum of all divisors of N

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, ...

Sloane's A000203, the sum of all divisors of N. More terms here

A080307 : Multiples of Fermat numbers

3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 21, 24, 25, 27, 30, 33, ...

Links to my notes on Fermat numbers.

A005408 : Odd numbers 2N+1

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, ...

Appears on the Narayana triangle page as a 3nd-order difference.

A171877 : An = An-1 + An-2 An-3 + An-4 An-5An-6

0, 0, 1, 1, 1, 1, 3, 5, 9, 25, 73, 313, 3263, 1502337, 278472902914281, 11984387434132924341157279996736444304839056033321, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

An = An-1 + An-2 An-3 + An-4An-5

0, 0, 1, 1, 1, 3, 5, 9, 25, 73, 423, 61297, 3814697357801, 38288777744833624093154249190851262684887027625, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1.

An = An-1 + An-2 An-3 + An-4 An-5An-6 + An-7 An-8An-9An-10

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 5, 9, 26, 75, 325, 3401, 1563053, 407212778591593, 1834324679277188341330884213675744831969260330864096, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1, and also uses the "lower hyper4" operator defined here.

An = An-1 + An-2 An-3 + An-4 An-5An-6 + An-7 An-8An-9An-10

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 5, 10, 27, 81, 367, 3805, 2733535, 16677181703796554, 1240970832262840567855391280367952509479536766933320366, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1, and also uses the "lower hyper4" operator defined here.

A094358 : Squarefree products of factors of Fermat numbers

1, 3, 5, 15, 17, 51, 85, 255, 257, 641, 771, 1285, 1923, 3205, 3855, 4369, 9615, 10897, 13107, 21845, 32691, 54485, 65535, 65537, 114689, 163455, 164737, 196611, ...

This sequence, Sloane's A094358, has its own page here.

I conjecture that this sequence contains all values of N such that 2N ≡ 1 mod N, and no other values.

Links to my page on the topic, lists me as author and mentions the conjecture linking the sequence to A023394.

A023394 : Prime Factors of Fermat Numbers

3, 5, 17, 257, 641, 65537, 114689, 274177, 319489, 974849, 2424833, 6700417, 13631489, 26017793, 45592577, 63766529, 167772161, 825753601, ...

This sequence has its own page here.

A050922 : Fermat Numbers, factorized

3, 5, 17, 257, 65537, 641, 6700417, 274177, 67280421310721, 59649589127497217, 5704689200685129054721, ...

Like A023394, this sequence links to my fermat numbers page.

A000215 : Fermat numbers 22N+1

3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, ...

This sequence has its own page here.

A000217 : Triangular numbers

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, ...

Appears as the 3rd diagonal of the Pascal triangle and the 2nd diagonal of the Narayana triangle.

An = An-1 + An-2 An-3 + An-4 An-5An-6 + An-7 An-8An-9An-10

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 3, 6, 11, 31, 101, 461, 5969, 54970924, 2566256166594610582, 62757193346815419996912506199334550962862239663434039815556137875, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1, and also uses the "lower hyper4" operator defined here.

MCS429697: floor(3N2/4)

0, 0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, 126, 147, 168, 192, 216, 243, 270, 300, 330, 363, 396, 432, 468, 507, 546, 588, 630, 675, 720, 768, 816, ...

A0 = 0; A1 = 0; AK+1 = AK-1 + 3 K

The even-numbered terms are the same as the three-quarter squares; the odd-numbered terms are one less.

A000740 : Restricted class of Necklaces, or Period-N mu-atoms in Mandelbrot Set

1, 1, 1, 3, 6, 15, 27, 63, 120, 252, 495, 1023, 2010, 4095, 8127, 16365, 32640, 65535, 130788, 262143, 523770, 1048509, 2096127, 4194303, 8386440, 16777200, 33550335, 67108608, 134209530, 268435455, 536854005, 1073741823, 2147450880, ...

This sequence is discussed here and here.

I submitted this to NJAS in early 2000, I am listed as having "proved" the connection to the Mandelbrot set (although the majority of the work had already been done by others who explored and described the complex dynamics of the iterated quadratic functions that define the Julia sets)

A005646 : Classifications of N Elements

1, 1, 1, 3, 6, 26, 122, 1015, 11847, 208914, 5236991, 184321511, ...

This is a very "hard" sequence to calculate. More on this page.

A000959 : Lucky numbers

1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303, ...

This sequence, like the primes, can be generated by a "sieve". There is a nice video about them by Numberphile: What is a lucky number?.

A005228 : Hofstadter Figure-Figure Sequence "R"

1, 3, 7, 12, 18, 26, 35, 45, 56, 69, 83, 98, 114, 131, 150, 170, 191, 213, 236, 260, 285, 312, 340, 369, 399, 430, 462, 495, 529, 565, 602, 640, 679, 719, 760, 802, 845, 889, 935, 982, 1030, 1079, ...

Appears in Gödel,_Escher,_Bach and defined recursively in terms of itself and its complement in the positive integers.

A077043 : "Three-Quarter Squares", or Hexagonal and "Semi-Hexagonal" Numbers

0, 1, 3, 7, 12, 19, 27, 37, 48, 61, 75, 91, 108, 127, 147, 169, 192, 217, 243, 271, 300, 331, 363, 397, 432, 469, 507, 547, 588, 631, 675, 721, 768, 817, 867, 919, 972, 1027, ...

A0 = 0; A1 = 1; AK+1 = AK-1 + 3 K

This is MCS1678, and should not be confused with A002061 or A000124. The even terms are A033428 and the odd terms are A003215. There is some discussion of this sequence in my entries for the numbers 27 and 61. Also compare to MCS429697.

A002061 or MCS62032 : Hogben's Centered Polygonal Numbers N2-N+1

1, 1, 3, 7, 13, 21, 31, 43, 57, 73, 91, 111, 133, 157, 183, 211, 241, 273, 307, 343, 381, 421, 463, 507, 553, 601, 651, 703, 757, 813, 871, 931, 993, 1057, 1123, 1191, 1261, 1333, 1407, 1483, ...

This is MCS62032, and should not be confused with A077043 or A000124. It gives a simple lower bound to the *kissing spheres" problem in N dimensions; this sequence has its own page here.

A136580 : Sums of alternating factorials A(N) = N! + (N-2)! + ... + 1.

1, 1, 3, 7, 27, 127, 747, 5167, 41067, 368047, 3669867, 40284847, 482671467, 6267305647, 87660962667, 1313941673647, 21010450850667, 357001369769647, 6423384156578667, 122002101778601647, 2439325392333218667, 51212944273488041647, 1126440053169940898667, 25903229683158464681647, 621574841786409380258667, 15537113273014144448681647, 403913035968392044964258667, 10904406563691366305216681647, 305292257647682252546468258667, 8852666400303393320848832681647, 265558152069838740888854948258667, 8231691320578226211046411712681647, 263396395085763368908106867108258667, ...

The terms are simply sums of factorials, for example 1+6+120=127 and 1+2+24+720=747. I like it because it starts with a few of my favorite numbers: 3, 7, 27 and 127. (See 715).

MCS8041809: Rilybeast Breeding Numbers

-1, 3, 7, 27, 143, 1011, 9111, 100235, 1303071, 19546083, 332283431, 6313385211, 132581089455, ...

This sequence has its own page here.

A000668 : Mersenne primes

3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, ...

These are the prime numbers expressible in the form 2p-1, where p is an integer. These p, called Mersenne exponents, are by necessity prime, but not all primes are Mersenne exponents. Here is a list of Mersenne primes.

A001566 : Previous term squared minus 2

3, 7, 47, 2207, 4870847, 23725150497407, 562882766124611619513723647, 316837008400094222150776738483768236006420971486980607, ...

This sequence is discussed here.

A082897 : Perfect totient numbers

3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, 5571, 6561, 8751, 15723, 19683, 36759, 46791, 59049, 65535, 140103, 177147, 208191, 441027, 531441, ...

This sequence has its own page here.

A073733 : Convergents to log210

1, 3, 10, 93, 196, 485, 2136, 13301, 28738, 42039, 70777, 254370, 325147, 6107016, 6432163, 44699994, 51132157, 146964308, 198096465, 345060773, 1578339557, 1923400330, ...

This sequence has its own page here.

A006876 : Period-N mu-molecules in Mandelbrot Set

1, 0, 1, 3, 11, 20, 57, 108, 240, 472, 1013, 1959, 4083, 8052, 16315, 32496, 65519, 130464, 262125, 523209, 1048353, 2095084, 4194281, 8384100, 16777120, ...

This sequence is discussed here and here.

OEIS entry links to the mu-ency title page.

A033428 or MCS1763 : 3n2

0, 3, 12, 27, 48, 75, 108, 147, 192, 243, 300, 363, 432, 507, 588, 675, 768, 867, 972, 1083, 1200, 1323, 1452, 1587, 1728, 1875, 2028, 2187, 2352, 2523, 2700, 2883, 3072, 3267, 3468, 3675, 3888, 4107, 4332, ...

These can be laid out geometrically as a "semi-hexagon": o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 3 o o o o 12 27 . . . which is easily seen to be three Lozenge or "diamond" shapes each containing n2, and fit together like a Triskelion. Together with the "true" hexagonal numbers A003215 they form sequence A077043.

A119906 : Largest number whose factorial is less than 1010N

3, 13, 69, 449, 3248, 25205, 205022, 1723507, 14842906, 130202808, 1158787577, 10433891463, 94851898540, 869200494599, 8019346203785, 74419210652835, 694100859679691, 6502464891216879, 61154108320430275, 577134533044522749, ...

I don't reference this sequence anywhere; I just think this sort of thing is cool.

A171883 (not yet in OEIS) : Mills primes, starting with 3

3, 29, 24391, 14510715208481, 3055388613462301256452407743005777548691, 28523273576637848665919896495441825882152454136941318837931307249186500390142888912756816872323385929633845346711703957, ...

Compare to A051254.

A097486 : Approximation of π10N by Iterating the Mandelbrot Set at -3/4+10-Ni

3, 33, 315, 3143, 31417, 314160, 3141593, 31415927, 314159266, 3141592655, 31415926537, 314159265359, 3141592653591, ...

There is more about this sequence in my discussion of Seahorse valley. My results do not agree precisely with OEIS; I get 31415927 for the N=7 term.

A006890 : Feigenbaum constant

4, 6, 6, 9, 2, 0, 1, 6, 0, 9, 1, 0, 2, 9, 9, 0, 6, 7, 1, 8, ...

My mu-ency page on the Feigenbaum constant is listed as a link.

A001358 : Semiprimes

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187, ...

These are numbers whose prime factorization has exactly two primes (counted with multiplicity). See also A014612.

A034189 : 2-colorings of 4-dimensional hypercube

1, 1, 4, 6, 19, 27, 50, 56, 74, 56, 50, 27, 19, 6, 4, 1, 1

This sequence shows up on my A005646 page.

An = An-1 + An-2 An-3 + An-4 An-5An-6 + An-7 An-8An-9An-10

0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 4, 7, 13, 43, 139, 727, 37918, 2698325206, 238838713275241145040397, ...

One of my accelerating sequences. The definition assumes the customary rule that 00 is 1, and also uses the "lower hyper4" operator defined here.

A013929 : Divisible by a square greater than 1

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, ...

In the factorization of these numbers, at least one prime factor has an exponent of at least 2.

A001694 : Powerful (or squarefull, square-full or 2-full) numbers: if a prime p divides n then p2 must also divide n

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, ...

In the factorization of these numbers, all prime factors have an exponent of at least 2.

A038109 : Divisible exactly by the square of a prime

4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, ...

In the factorization of these numbers, at least one prime factor has an exponent of exactly 2.

Exactly one prime factor has exponent exactly 2

4, 9, 12, 18, 20, 25, 28, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 121, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 188, 198, 200, 204, 207, 212, 220, ...

In the factorization of these numbers, exactly one prime factor has an exponent of exactly 2.

A000290 : Squares AN=N2

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, ...

In the factorization of these numbers, all prime factors have an exponent that is a multiple of 2.

Appears on the Narayana triangle page as a 2nd-order difference.

A019298 : Smallest fruit pyramid of height N

0, 1, 4, 11, 23, 42, 69, 106, 154, 215, 290, 381, 489, 616, 763, 932, 1124, 1341, 1584, 1855, 2155, 2486, 2849, 3246, 3678, 4147, 4654, 5201, 5789, 6420, 7095, 7816, 8584, 9401, 10268, 11187, 12159, 13186, 14269, 15410, ...

This is MCS12904546, its first differences are A077043. An iterative formula (recurrence relation) is:

A0 = 0; A1 = 1; A2 = 4; AK+1 = AK + AK-1 - AK-2 + 3 K

A003010 : Lucas-Lehmer sequence for Mersenne prime testing.

4, 14, 194, 37634, 1416317954, 2005956546822746114, 4023861667741036022825635656102100994, 16191462721115671781777559070120513664958590125499158514329308740975788034, ...

This is the sequence used in testing a candidate Mersenne number 2N-1 to determine if it is prime. 2N-1 is a prime if and only if it divides AN-2 (and otherwise it is composite). Since 23-1 is a divisor of A1=14, 23-1=7 is a prime. Similarly, since 24-1 is not a divisor of A2=194, 24-1=15 is not a prime. And since 25-1 is a divisor of A3=37634, 25-1=31 is a prime.

A002109 : Hyperfactorials: Productk = 1..n kk.

1, 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, 55696437941726556979200000, 21577941222941856209168026828800000, 215779412229418562091680268288000000000000000, 61564384586635053951550731889313964883968000000000000000, ...

A53015 : NN(N-1) prefixed by an initial 0

0, 1, 4, 19683, 340282366920938463463374607431768211456

This is also A089210 extended back to N=0, and discussed on my hyper4 page.

A098403 : Area of Mandelbrot set

1, 5, 0, 6, 5, 9, 1, ...

My mu-ency page on the Area of the Mandelbrot Set is listed as a link.

A078335 : Largest real root of eX = Gamma(X)

5, 2, 9, 0, 3, 1, 6, 0, 9, 3, 1, 1, 9, 7, 7, 0, 7, 1, 0, 7, 2, 2, 2, 2, 5, 8, 1, 8, 6, 3, 1, 1, 7, 2, 7, 4, 7, 9, 9, 9, 8, 2, 0, 1, 8, 8, 3, 0, 7, 5, 4, 4, 2, 9, 2, 3, 9, 4, ...

This is simply the digits of the number 5.2903160931..., above which the Gamma function exceeds eX. This sequence links to my numbers page.

A008587 : Multiples of five

0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, ...

The most fun row of the multiplication table to say aloud (see Schoolhouse Rock); also appears on the Narayana triangle page as a 5th-order difference.

A034190 : 2-colorings of 5-dimensional hypercube

1, 1, 5, 10, 47, 131, 472, 1326, 3779, 9013, 19963, 38073, 65664, 98804, 133576, 158658, 169112, 158658, 133576, 98804, 65664, 38073, 19963, 9013, 3779, 1326, 472, 131, 47, 10, 5, 1, 1

This sequence shows up on my A005646 page.

A000330 : Square pyramidal numbers

0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416, 11440, 12529, 13685, 14910, 16206, 17575, 19019, 20540, 22140, 23821, 25585, 27434, 29370, ...

This sequence is discussed on the Narayana triangle page.

A001622 : Digits of {phi} (phi)

1, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, ...

A078333 : sqrt(2)sqrt(2)

1, 6, 3, 2, 5, 2, 6, 9, 1, 9, 4, 3, 8, 1, 5, 2, 8, 4, 4, 7, 7, 3, 4, 9, 5, 3, 8, 1, 0, 2, 4, 7, 1, 9, 6, 0, 2, 0, 7, 9, 1, 0, 8, 8, 5, 7, 0, 5, 3, 1, 1, 4, 1, 1, 7, 2, 4, 7, ...

This is simply the digits of the number 1.632526919..., which can be used to prove an interesting thing about irrational numbers; see here. The OEIS entry for this sequence links to my numbers page.

A006881 : Product of two distinct primes

6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, ...

I am listed as a co-author, I probably called attention to it in connection with A055233.

A000384 : Hexagonal Numbers N(2N-1)

0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278, 2415, 2556, 2701, 2850, ...

This is MCS7066. Compare to A000566 and A003215.

A005891 : Centered pentagonal numbers

1, 6, 16, 31, 51, 76, 106, 141, 181, 226, 276, 331, 391, 456, 526, 601, 681, 766, 856, 951, 1051, 1156, 1266, 1381, 1501, 1626, 1756, 1891, 2031, 2176, 2326, 2481, 2641, 2806, 2976, ...

This is also the 4th-order difference of one of the diagonals of the Narayana triangle and is discussed more fully here.

A019296 : Values of N for which eπ sqrt(N) is nearly an integer

-1, 0, 6, 17, 18, 22, 25, 37, 43, 58, 59, 67, 74, 103, 148, 149, 163, 164, 177, 205, 223, 226, 232, 267, 268, 326, 359, 386, 522, 566, 630, 638, 652, 719, 790, 792, 928, 940, 986, 1005, 1014, ...

For the first two terms e^{π sqrt(N) is exactly an integer: e^(πi} = -1 and e0 = 1.

This sequence, Sloane's A019296, and several related sequences have their own page here.

A002415 : 4-dimensional pyramidal numbers

0, 0, 1, 6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736, 111265, 124950, 139860, 156066, 173641, ...

This is the 3rd diagonal of the Narayana triangle.

A000396 : Perfect numbers

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, ...

Connected to the unsolved problem of Mersenne primes; a longer list is here.

MCS496267 : A puzzle sequence

0, 6, 42, 156, 420, 930, 1806, 3192, 5256, 8190, 12210, 17556, 24492, 33306, 44310, 57840, 74256, 93942, 117306, 144780, 176820, 213906, 256542, 305256, ...

This is MCS496267, and is defined by the recurrence relation A0 = 0; AK+1 = AK + 4 K3 + 2 K. It is the "simplest" recurrence-defined solution to a puzzle sequence given to the seqfan list by Terry Stickel in July 2010.

With different offsets the sequence appears in OEIS as A176780 and as A169938, and possibly also A082986. It is also equivalent to A002378(A002378(N)).

The sequence comes up in this discussion (in French, and which I have not examined in detail).

A006878 : Successive record values of Np for Collatz 3X+1 iteration

1, 7, 8, 16, 19, 20, 23, 111, 112, 115, 118, 121, 124, 127, 130, 143, 144, 170, 178, 181, 182, 208, 216, 237, 261, 267, 275, 278, 281, 307, 310, 323, 339, 350, 353, 374, 382, 385, 442, 448, 469, ...

This sequence is discussed here.

I calculated this sequence on an Apple ][ shortly after the publication of Gödel,_Escher,_Bach and submitted it to N.J.A. Sloane in 1994.

A162018 : 2N ≠ 22N mod N

7, 9, 11, 13, 19, 21, 22, 23, 25, 27, 29, 31, 33, 35, 37, 38, 39, 41, 45, 47, 49, 50, 53, 54, 55, 57, 59, 61, 62, 63, 65, 66, 67, 69, 71, 73, 74, 75, 77, 79, 81, 82, 83, 86, 87, 89, 91, 92, 93, 94, ...

This sequence is discussed here.

A094534 : Centered Hexamorphic Numbers

1, 7, 17, 51, 67, 167, 251, 417, 501, 667, 751, 917, 1251, 1667, 5001, 5417, 6251, 6667, 10417, 16667, 50001, 56251, 60417, 66667, 166667, 260417, 406251, 500001, 666667, 760417, ...

This sequence, Sloane's A094534, has its own page here.

Lists me as author, and links to my A094534 page.

A000566 : Heptagonal Numbers N(5N-3)/2

0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, ...

This is MCS15655. See Heptagonal number, and compare to A000384 and A069099.

A003215 : Centered Hexagonal Numbers 3N(N+1)+1

1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817, 919, 1027, 1141, 1261, 1387, 1519, 1657, 1801, 1951, 2107, 2269, 2437, 2611, 2791, 2977, 3169, 3367, 3571, 3781, 3997, 4219, 4447, 4681, 4921, 5167, 5419, ...

This is MCS496592. Compare to A000566 and A094534.

A147587 : 14N+7, or odd numbers times seven.

7, 21, 35, 49, 63, 77, 91, 105, 119, 133, 147, 161, 175, 189, 203, 217, 231, 245, 259, 273, 287, 301, 315, ...

This sequence is the 7th-order difference of one of the diagonals of the Narayana triangle.

A004068 : Number of atoms in dodecahedron with n shells

0, 1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, 24831, 27312, 29953, 32759, 35735, 38886, 42217, 45733, 49439, ...

This is also the 3rd-order difference of one of the diagonals of the Narayana triangle and is discussed more fully here.

Kaprekar numbers for 4th powers

1, 7, 45, 55, 67, 100, 433, 4950, 5050, 38212, 65068, 190576, 295075, 299035, 310024, 336700, 343333, 394615, 414558, 433566, 448228, 450550, 467236, 475497, 476191, 486486, ...

This sequence is discussed here. (Submitted as A171493; not yet approved)

A060809 : Kaprekar numbers for 3rd powers, allowing intermediate result of fewer digits

1, 8, 10, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 27100, 44443, 55556, 60434, 77778, 143857, 208494, 226071, 279720, 313390, 324675, 329967, 346060, ...

Cited in OEIS as being "erroneous", but actually just defined differently. Discussed here.

A014612 : "triprimes" or "3-almost primes"

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 148, 153, 154, 164, 165, 170, 171, 172, 174, 175, 182, 186, 188, 190, 195, 207, 212, 222, 230, 231, 236, 238, 242, 244, ...

These are numbers whose prime factorization has exactly three primes (counted with multiplicity). See also A001358.

A171607 : Expressible as A×BA in a nontrivial way

8, 18, 24, 32, 50, 64, 72, 81, 98, 128, 160, 162, 192, 200, 242, 288, 324, 338, 375, 384, 392, 450, 512, 578, 648, 722, 800, 882, 896, 968, 1024, 1029, 1058, 1152, ...

This sequence is discussed here.

A069099 : Centered Heptagonal Numbers

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 386, 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, ...

This is MCS1986496, with the formula A0 = 1; AK+1 = AK + 7 K. See Centered heptagonal number, and compare to A000566 and A003215.

A000578 : The cubes N3

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 46656, 50653, 54872, 59319, 64000, ...

A006322 : 4-dimensional hyperpyramid based on pentagonal pyramids

1, 8, 31, 85, 190, 371, 658, 1086, 1695, 2530, 3641, 5083, 6916, 9205, 12020, 15436, 19533, 24396, 30115, 36785, 44506, 53383, 63526, 75050, 88075, 102726, 119133, 137431, 157760, 180265, 205096, 232408, 262361, 295120, 330855, ...

This is also the 2nd-order difference of one of the diagonals of the Narayana triangle and is discussed more fully here.

A006887 : Kaprekar numbers for 3rd powers, omitting 10x

1, 8, 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 27100, 44443, 55556, 60434, 77778, 143857, 208494, 226071, 279720, 313390, 324675, 329967, 346060, 368928, 395604, 422577, 427868, 461539, 472823, 478115, 488214, 494208, ...

This sequence is discussed here.

I submitted this sequence to Sloane sometime in the early 1990's, at the same time as the normal (square) Kaprekar numbers.

A005919 : 7N2 + 2

1, 9, 30, 65, 114, 177, 254, 345, 450, 569, 702, 849, 1010, 1185, 1374, 1577, 1794, 2025, 2270, 2529, 2802, 3089, 3390, 3705, 4034, 4377, 4734, 5105, 5490, 5889, 6302, 6729, 7170, 7625, 8094, 8577, 9074, 9585, 10110, 10649, 11202, 11769, ...

1st term given in the OEIS entry does not agree with the formula. This sequence is the 6th-order difference of one of the diagonals of the Narayana triangle.

A006414 : Number of nonseparable toroidal tree-rooted maps on n nodes

1, 9, 40, 125, 315, 686, 1344, 2430, 4125, 6655, 10296, 15379, 22295, 31500, 43520, 58956, 78489, 102885, 133000, 169785, 214291, 267674, 331200, 406250, 494325, 597051, 716184, 853615, 1011375, 1191640, 1396736, 1629144, 1891505, 2186625, 2517480, 2887221, ...

This is also the 1st-order difference of one of the diagonals of the Narayana triangle and is discussed more fully here.

A006886 : Kaprekar numbers for squares, allowing leading 0

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, ...

There are other definitions of Kaprekar numbers, including A053816

This sequence is discussed in more detail here.

I am listed as author. I found the Kaprekar numbers in Wells and sent them to Sloane with many other suggestions, in the early 1990's.

A053816 : Kaprekar numbers for squares

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, ...

As with A006886, I am listed as author.

In Sloane's 1995 book this sequence is called "A6886", but it was later decided to add the terms 4879 and 5292 because that is more fitting to Kaprekar's definition.

This sequence is discussed in more detail here.

A055233 : N = P + ... + Q, P and Q are smallest and largest prime factors of N

10, 39, 155, 371, 2935561623745, ...

454539357304421 is a member but it is yet unknown if any terms exist between 2935561623745 and 454539357304421.

I was the first to give an extension to the "39-property", the 454539357304421 term, on March 7, 2000. See this article in sci.math.

Sloane also links to a primepuzzles.net article. If you look at the dates, it appears that Jud McCranie wrote on January 7th, March 7th and July 7th 2000. However if you look here or on any of a number of other Prime Puzzles articles, you can see that the dates are given as Day/Month/Year — so in article 98, Jud McCranie actually write on the 1st, 3rd and 7th of July.

I consider the problem of N = P Q = P + ... + Q to be more interesting, but Sloane does not yet have an entry for that. If he did, 454539357304421 would be the next term after 10, 39, 155 and 371.

A063490 : (2N-1)(7N2-7N+6)/6

1, 10, 40, 105, 219, 396, 650, 995, 1445, 2014, 2716, 3565, 4575, 5760, 7134, 8711, 10505, 12530, 14800, 17329, 20131, 23220, 26610, 30315, 34349, 38726, 43460, 48565, 54055, 59944, 66246, 72975, 80145, 87770, 95864, 104441, 113515, ...

This sequence is the 5th-order difference of one of the diagonals of the Narayana triangle.

A006542 : C(n,3)C(n-1,3)/4

1, 10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626, 3708810, 4562425, 5573800, 6765440, 8162176, ...

This sequence is the 4th diagonal of the Narayana triangle, and also has its own page here.

Kaprekar numbers for 5th powers

1, 10, 1000, 7776, 27100, 73440, 95120, 500499, 505791, 540539, 598697, 665335, 697598, 732347, 7607610, 37944478, 46945205, 54995500, 55216205, 56607166, ...

This sequence is discussed here. (Submitted as A171500, not yet approved)

A005150 : The 'Look and Say' sequence: each term 'describes' the previosus term.

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, 11131221133112132113212221, ...

The first term is 1, which is "one 1" or "1 1". The second term is 11 which is "two ones" or "2 1". The third term is 21 which is "one 2, one 1" or "1 2 1 1". The 4th term is 1211... No term has a digit bigger than 3.

MCS6990239: A0=0; A1=1; AN+1=2AN-AN-1+7N2+2

0, 1, 11, 51, 156, 375, 771, 1421, 2416, 3861, 5875, 8591, 12156, 16731, 22491, 29625, 38336, 48841, 61371, 76171, 93500, 113631, 136851, 163461, 193776, 228125, ...

This sequence is the 4th-order difference of one of the diagonals of the Narayana triangle.

A085463 : N(N+1)(2N+1)(7N2+7N+6)/120

1, 12, 63, 219, 594, 1365, 2786, 5202, 9063, 14938, 23529, 35685, 52416, 74907, 104532, 142868, 191709, 253080, 329251, 422751, 536382, 673233, 836694, 1030470, 1258595, 1525446, 1835757, 2194633, 2607564, 3080439, 3619560, 4231656, ...

This sequence is the 3rd-order difference of one of the diagonals of the Narayana triangle.

A114244 : (N+1)(N+2)2(N+3)(7N^2+28N+30)/360

1, 13, 76, 295, 889, 2254, 5040, 10242, 19305, 34243, 57772, 93457, 145873, 220780, 325312, 468180, 659889, 912969, 1242220, 1664971, 2201353, 2874586, 3711280, 4741750, 6000345, 7525791, 9361548, 11556181, 14163745, 17244184, 20863744, ...

This sequence is the 2nd-order difference of one of the diagonals of the Narayana triangle.

A114242 : (N+1)(N+2)2(N+3)2(N+4)(2N+5)/720

1, 14, 90, 385, 1274, 3528, 8568, 18810, 38115, 72358, 130130, 223587, 369460, 590240, 915552, 1383732, 2043621, 2956590, 4198810, 5863781, 8065134, 10939720, 14651000, 19392750, 25393095, 32918886, 42280434, 53836615, 68000360, ...

This sequence is the 1st-order difference of one of the diagonals of the Narayana triangle.

A006857 : Restricted class of permutations

1, 15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, 26883780, 37823500, 52474500, 71867250, 97260345, 130179231, 172459665, 226296280, ...

This sequence is the 5th diagonal of the Narayana triangle.

A054670 : Laurent series for mapping onto boundary of Mandelbrot set (Numerators)

-1, 1, -1, 15, 0, -47, -1, 987, 0, -3673, 1, -61029, 0, -689455, -21, 59250963, 0, -164712949, 39, ...

Lists me as author, and links to my mu-ency page on the Laurent series for the Maple code.

2nd factorials, or xxx hyperfactorials: Productk = 1..n kkk

1, 1, 16, 122009559759792, 1635880742876144370140651613365618678672431446259953350330107904905235944459582740143020318091028072360877064844748388419544694286698651060181531410733823915504311468032, ...

Notes by Bill Gosper, from approximately 1978, describe a series of factorial functions. The series begins with the normal factorials, and each has an analytic extension (which is the Gamma function in the case of the normal factorials). In this document the normal factorials are called the "0th factorials". The "hyperfactorials" A002109 (1, 1, 4, 108, 27648, ...; listed above) are called the "1st factorials"; and this sequence is the "2nd factorials".

A008884 : Collatz 3X+1 iteration starting with 27

27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 4, 2, 1, ...

This sequence is discussed more fully here.

A064224 : Has more than one representation as the product of consecutive integers > 1

120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, ...

This sequence has its own page here.

Gives my extension from 20070813; still waiting for my more recent extension to be added.

Markov numbers that are neither Fibonacci nor Pell

(not in OEIS)

194, 433, 1325, 2897, 6466, 7561, 9077, 14701, 37666, 43261, 51641, 62210, 96557, 135137, 294685, 426389, 499393, 646018, 925765, ...

A171606 : Expressible as A×BA in two or more different ways

648, 2048, 4608, 5184, 41472, 52488, 472392, 500000, 524288, 2654208, 3125000, 4718592, 10125000, 13436928, 21233664, 30233088, 46118408, 76236552, 134217728, 169869312, 344373768, 402653184, 512000000, 648000000, 737894528, 800000000, 838860800, 922640625, 1147971528, 1207959552, 1714871048, 1934917632, 2754990144, 3127772232, 3439853568, 4879139328, 6525845768, 6973568802, 7381125000, ...

This sequence is discussed here.


If this sort of thing is of interest, you should also take a look at my comprehensive list of sequences generated by "classical" formulas like AN = 2N+1 and A0=0; A1=1; AN+1 = AN-1+N.

This page is meant to counteract the forces of Munafo's Law of Mathematical Discourse. If you see room for improvement, let me know!


Longer Lists of Sequence Terms

A000203: Sum of all divisors of N

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 124, 140, 96, 168, 80, 186, 121, 126, 84, 224, 108, 132, 120, 180, 90, 234, 112, 168, 128, 144, 120, 252, 98, 171, 156, 217, 102, 216, 104, 210, 192, 162, 108, 280, 110, 216, 152, 248, 114, 240, 144, 210, 182, 180, 144, 360, 133, 186, 168, 224, 156, 312, 128, 255, 176, 252, 132, 336, 160, 204, 240, 270, 138, 288, 140, 336, 192, 216, 168, 403, 180, 222, 228, 266, 150, 372, 152, 300, 234, 288, 192, 392, 158, 240, 216, 378, 192, 363, 164, 294, 288, 252, 168, 480, 183, 324, 260, 308, 174, 360, 248, 372, 240, 270, 180, 546, 182, 336, 248, 360, 228, 384, 216, 336, 320, 360, 192, 508, 194, 294, 336, 399, 198, 468, 200, 465, 272, 306, 240, 504, 252, 312, 312, 434, 240, 576, 212, 378, 288, 324, 264, 600, 256, 330, 296, 504, 252, 456, 224, 504, 403, 342, 228, 560, 230, 432, 384, 450, 234, 546, 288, 420, 320, 432, 240, 744, 242, 399, 364, 434, 342, 504, 280, 480, 336, 468, 252, 728, 288, 384, 432, 511, 258, 528, 304, 588, 390, 396, 264, 720, 324, 480, 360, 476, 270, 720, 272, 558, 448, 414, 372, 672, 278, 420, 416, 720, 282, 576, 284, 504, 480, 504, 336, 819, 307, 540, 392, 518, 294, 684, 360, 570, 480, 450, 336, 868, 352, 456, 408, 620, 372, 702, 308, 672, 416, 576, 312, 840, 314, 474, 624, 560, 318, 648, 360, 762, 432, 576, 360, 847, 434, 492, 440, 630, 384, 864, 332, 588, 494, 504, 408, 992, 338, 549, 456, 756, 384, 780, 400, 660, 576, 522, 348, 840, 350, 744, 560, 756, 354, 720, 432, 630, 576, 540, 360, 1170, 381, 546, 532, 784, 444, 744, 368, 744, 546, 684, 432, 896, 374, 648, 624, 720, 420, 960, 380, 840, 512, 576, 384, 1020, 576, 582, 572, 686, 390, 1008, 432, 855, 528, 594, 480, 1092, 398, 600, 640, 961, 402, 816, 448, 714, 726, 720, 456, 1080, 410, 756, 552, 728, 480, 936, 504, 882, 560, 720, 420, 1344, 422, 636, 624, 810, 558, 864, 496, 756, 672, 792, 432, 1240, 434, 768, 720, 770, 480, 888, 440, 1080, 741, 756, 444, 1064, 540, 672, 600, 1016, 450, 1209, 504, 798, 608, 684, 672, 1200, 458, 690, 720, 1008, 462, 1152, 464, 930, 768, 702, 468, 1274, 544, 864, 632, 900, 528, 960, 620, 1008, 702, 720, 480, 1512, 532, 726, 768, 931, 588, 1092, 488, 930, 656, 1026, 492, 1176, 540, 840, 936, 992, 576, 1008, 500, 1092, 672, 756, 504, 1560, 612, 864, 732, 896, 510, 1296, 592, 1023, 800, 774, 624, 1232, 576, 912, 696, 1260, 522, 1170, 524, 924, 992, 792, 576, 1488, 553, 972, 780, 1120, 588, 1080, 648, 1020, 720, 810, 684, 1680, 542, 816, 728, 1134, 660, 1344, 548, 966, 806, 1116, 600, 1440, 640, 834, 912, 980, 558, 1248, 616, 1488, 864, 846, 564, 1344, 684, 852, 968, 1080, 570, 1440, 572, 1176, 768, 1008, 744, 1651, 578, 921, 776, 1260, 672, 1176, 648, 1110, 1092, 882, 588, 1596, 640, 1080, 792, 1178, 594, 1440, 864, 1050, 800, 1008, 600, 1860, 602, 1056, 884, 1064, 798, 1224, 608, 1260, 960, 1116, 672, 1638, 614, 924, 1008, 1440, 618, 1248, 620, 1344, 960, 936, 720, 1736, 781, 942, 960, 1106, 684, 1872, 632, 1200, 848, 954, 768, 1512, 798, 1080, 936, 1530, 642, 1296, 644, 1344, 1056, 1080, 648, 1815, 720, 1302, 1024, 1148, 654, 1320, 792, 1302, 962, 1152, 660, 2016, 662, 996, 1008, 1260, 960, 1482, 720, 1176, 896, 1224, 744, 2016, 674, 1014, 1240, 1281, 678, 1368, 784, 1620, 912, 1152, 684, 1820, 828, 1200, 920, 1364, 756, 1728, 692, 1218, 1248, 1044, 840, 1800, 756, 1050, 936, 1736, 702, 1680, 760, 1524, 1152, 1062, 816, 1680, 710, 1296, 1040, 1350, 768, 1728, 1008, 1260, 960, 1080, 720, 2418, 832, 1143, 968, 1274, 930, 1596, 728, 1680, 1093, 1332, 792, 1736, 734, 1104, 1368, 1512, 816, 1638, 740, 1596, 1120, 1296, 744, 1920, 900, 1122, 1092, 1512, 864, 1872, 752, 1488, 1008, 1260, 912, 2240, 758, 1140, 1152, 1800, 762, 1536, 880, 1344, 1404, 1152, 840, 2044, 770, 1728, 1032, 1358, 774, 1716, 992, 1470, 1216, 1170, 840, 2352, 864, 1296, 1200, 1767, 948, 1584, 788, 1386, 1056, 1440, 912, 2340, 868, 1194, 1296, 1400, 798, 1920, 864, 1953, 1170, 1206, 888, 1904, 1152, 1344, 1080, 1530, 810, 2178, 812, 1680, 1088, 1368, 984, 2232, 880, 1230, 1456, 1764, 822, 1656, 824, 1560, 1488, 1440, 828, 2184, 830, 1512, 1112, 1778, 1026, 1680, 1008, 1680, 1280, 1260, 840, 2880, 871, 1266, 1128, 1484, 1098, 1872, 1064, 1674, 1136, 1674, 912, 2016, 854, 1488, 1560, 1620, 858, 2016, 860, 1848, 1344, 1296, 864, 2520, 1044, 1302, 1228, 1792, 960, 2160, 952, 1650, 1274, 1440, 1248, 2072, 878, 1320, 1176, 2232, 882, 2223, 884, 1764, 1440, 1332, 888, 2280, 1024, 1620, 1452, 1568, 960, 1800, 1080, 2040, 1344, 1350, 960, 2821, 972, 1512, 1408, 1710, 1092, 1824, 908, 1596, 1326, 2016, 912, 2480, 1008, 1374, 1488, 1610, 1056, 2160, 920, 2160, 1232, 1386, 1008, 2688, 1178, 1392, 1352, 1890, 930, 2304, 1140, 1638, 1248, 1404, 1296, 2730, 938, 1632, 1256, 2016, 942, 1896, 1008, 1860, 1920, 1584, 948, 2240, 1036, 1860, 1272, 2160, 954, 2106, 1152, 1680, 1440, 1440, 1104, 3048, 993, 1596, 1404, 1694, 1164, 2304, 968, 1995, 1440, 1764, 972, 2548, 1120, 1464, 1736, 1922, 978, 1968, 1080, 2394, 1430, 1476, 984, 2520, 1188, 1620, 1536, 1960, 1056, 2808, 992, 2016, 1328, 1728, 1200, 2352, 998, 1500, 1520, ...

A000041: Partitions

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226, 239943, 281589, 329931, 386155, 451276, 526823, 614154, 715220, 831820, 966467, 1121505, 1300156, 1505499, 1741630, 2012558, 2323520, 2679689, 3087735, 3554345, 4087968, 4697205, 5392783, 6185689, 7089500, 8118264, 9289091, 10619863, 12132164, 13848650, 15796476, 18004327, 20506255, 23338469, 26543660, 30167357, 34262962, 38887673, 44108109, 49995925, 56634173, 64112359, 72533807, 82010177, 92669720, 104651419, 118114304, 133230930, 150198136, 169229875, 190569292, 214481126, 241265379, 271248950, 304801365, 342325709, 384276336, 431149389, 483502844, 541946240, 607163746, 679903203, 761002156, 851376628, 952050665, 1064144451, 1188908248, 1327710076, 1482074143, 1653668665, 1844349560, 2056148051, 2291320912, 2552338241, 2841940500, 3163127352, 3519222692, 3913864295, 4351078600, 4835271870, 5371315400, 5964539504, 6620830889, 7346629512, 8149040695, 9035836076, 10015581680, 11097645016, 12292341831, 13610949895, 15065878135, 16670689208, 18440293320, 20390982757, 22540654445, 24908858009, 27517052599, 30388671978, 33549419497, 37027355200, 40853235313, 45060624582, 49686288421, 54770336324, 60356673280, 66493182097, 73232243759, 80630964769, 88751778802, 97662728555, 107438159466, 118159068427, 129913904637, 142798995930, 156919475295, 172389800255, 189334822579, 207890420102, 228204732751, 250438925115, 274768617130, 301384802048, 330495499613, 362326859895, 397125074750, 435157697830, 476715857290, 522115831195, 571701605655, 625846753120, 684957390936, 749474411781, 819876908323, 896684817527, 980462880430, 1071823774337, 1171432692373, 1280011042268, 1398341745571, 1527273599625, 1667727404093, 1820701100652, 1987276856363, 2168627105469, 2366022741845, 2580840212973, 2814570987591, 3068829878530, 3345365983698, 3646072432125, 3972999029388, 4328363658647, 4714566886083, 5134205287973, 5590088317495, 6085253859260, 6622987708040, 7206841706490, 7840656226137, 8528581302375, 9275102575355, 10085065885767, 10963707205259, 11916681236278, 12950095925895, 14070545699287, 15285151248481, 16601598107914, 18028182516671, 19573856161145, 21248279009367, 23061871173849, 25025873760111, 27152408925615, 29454549941750, 31946390696157, 34643126322519, 37561133582570, 40718063627362, 44132934884255, 47826239745920, 51820051838712, 56138148670947, 60806135438329, 65851585970275, 71304185514919, 77195892663512, 83561103925871, 90436839668817, 97862933703585, 105882246722733, 114540884553038, 123888443077259, 133978259344888, 144867692496445, 156618412527946, 169296722391554, 182973889854026, 197726516681672, 213636919820625, 230793554364681, 249291451168559, 269232701252579, 290726957916112, 313891991306665, 338854264248680, 365749566870782, 394723676655357, 425933084409356, 459545750448675, 495741934760846, 534715062908609, 576672674947168, 621837416509615, 670448123060170, 722760953690372, 779050629562167, 839611730366814, 904760108316360, 974834369944625, 1050197489931117, 1131238503938606, 1218374349844333, 1312051800816215, 1412749565173450, 1520980492851175, 1637293969337171, 1762278433057269, 1896564103591584, 2040825852575075, 2195786311682516, 2362219145337711, 2540952590045698, 2732873183547535, 2938929793929555, 3160137867148997, 3397584011986773, 3652430836071053, 3925922161489422, 4219388528587095, 4534253126900886, 4872038056472084, 5234371069753672, 5622992691950605, 6039763882095515, 6486674127079088, 6965850144195831, 7479565078510584, 8030248384943040, 8620496275465025, 9253082936723602, 9930972392403501, 10657331232548839, 11435542077822104, 12269218019229465, 13162217895057704, 14118662665280005, 15142952738857194, 16239786535829663, 17414180133147295, 18671488299600364, 20017426762576945, 21458096037352891, 23000006655487337, 24650106150830490, 26415807633566326, 28305020340996003, 30326181989842964, 32488293351466654, 34800954869440830, 37274405776748077, 39919565526999991, 42748078035954696, 45772358543578028, 49005643635237875, 52462044228828641, 56156602112874289, ...

A000045: Fibonacci numbers

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, 2971215073, 4807526976, 7778742049, 12586269025, 20365011074, 32951280099, 53316291173, 86267571272, 139583862445, 225851433717, 365435296162, 591286729879, 956722026041, 1548008755920, 2504730781961, 4052739537881, 6557470319842, 10610209857723, 17167680177565, 27777890035288, 44945570212853, 72723460248141, 117669030460994, 190392490709135, 308061521170129, 498454011879264, 806515533049393, 1304969544928657, 2111485077978050, 3416454622906707, 5527939700884757, 8944394323791464, 14472334024676221, 23416728348467685, 37889062373143906, 61305790721611591, 99194853094755497, 160500643816367088, 259695496911122585, 420196140727489673, 679891637638612258, 1100087778366101931, 1779979416004714189, 2880067194370816120, 4660046610375530309, 7540113804746346429, 12200160415121876738, 19740274219868223167, 31940434634990099905, 51680708854858323072, 83621143489848422977, 135301852344706746049, 218922995834555169026, 354224848179261915075, 573147844013817084101, 927372692193078999176, 1500520536206896083277, 2427893228399975082453, 3928413764606871165730, 6356306993006846248183, 10284720757613717413913, 16641027750620563662096, 26925748508234281076009, 43566776258854844738105, 70492524767089125814114, 114059301025943970552219, 184551825793033096366333, 298611126818977066918552, 483162952612010163284885, 781774079430987230203437, 1264937032042997393488322, 2046711111473984623691759, 3311648143516982017180081, 5358359254990966640871840, 8670007398507948658051921, 14028366653498915298923761, 22698374052006863956975682, 36726740705505779255899443, 59425114757512643212875125, 96151855463018422468774568, 155576970220531065681649693, 251728825683549488150424261, 407305795904080553832073954, 659034621587630041982498215, 1066340417491710595814572169, 1725375039079340637797070384, 2791715456571051233611642553, 4517090495650391871408712937, 7308805952221443105020355490, 11825896447871834976429068427, 19134702400093278081449423917, 30960598847965113057878492344, 50095301248058391139327916261, 81055900096023504197206408605, 131151201344081895336534324866, 212207101440105399533740733471, 343358302784187294870275058337, 555565404224292694404015791808, 898923707008479989274290850145, 1454489111232772683678306641953, 2353412818241252672952597492098, 3807901929474025356630904134051, 6161314747715278029583501626149, 9969216677189303386214405760200, 16130531424904581415797907386349, 26099748102093884802012313146549, 42230279526998466217810220532898, 68330027629092351019822533679447, 110560307156090817237632754212345, 178890334785183168257455287891792, 289450641941273985495088042104137, 468340976726457153752543329995929, 757791618667731139247631372100066, 1226132595394188293000174702095995, 1983924214061919432247806074196061, 3210056809456107725247980776292056, 5193981023518027157495786850488117, 8404037832974134882743767626780173, 13598018856492162040239554477268290, 22002056689466296922983322104048463, 35600075545958458963222876581316753, 57602132235424755886206198685365216, 93202207781383214849429075266681969, 150804340016807970735635273952047185, 244006547798191185585064349218729154, 394810887814999156320699623170776339, 638817435613190341905763972389505493, 1033628323428189498226463595560281832, 1672445759041379840132227567949787325, 2706074082469569338358691163510069157, 4378519841510949178490918731459856482, 7084593923980518516849609894969925639, 11463113765491467695340528626429782121, 18547707689471986212190138521399707760, 30010821454963453907530667147829489881, 48558529144435440119720805669229197641, 78569350599398894027251472817058687522, 127127879743834334146972278486287885163, ...

A001013: Jordan-Polya numbers or "polyfactorials"

1, 2, 4, 6, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2304, 2592, 2880, 3072, 3456, 3840, 4096, 4320, 4608, 5040, 5184, 5760, 6144, 6912, 7680, 7776, 8192, 8640, 9216, 10080, 10368, 11520, 12288, 13824, 14400, 15360, 15552, 16384, 17280, 18432, 20160, 20736, 23040, 24576, 25920, 27648, 28800, 30240, 30720, 31104, 32768, 34560, 36864, 40320, 41472, 46080, 46656, 49152, 51840, 55296, 57600, 60480, 61440, 62208, 65536, 69120, 73728, 80640, 82944, 86400, 92160, 93312, 98304, 103680, 110592, 115200, 120960, 122880, 124416, 131072, 138240, 147456, 155520, 161280, 165888, 172800, 181440, 184320, 186624, 196608, 207360, 221184, 230400, 241920, 245760, 248832, 262144, 276480, 279936, 294912, 311040, 322560, 331776, 345600, 362880, 368640, 373248, 393216, 414720, 442368, 460800, 483840, 491520, 497664, 518400, 524288, 552960, 559872, 589824, 604800, 622080, 645120, 663552, 691200, 725760, 737280, 746496, 786432, 829440, 884736, 921600, 933120, 967680, 983040, 995328, 1036800, 1048576, 1088640, 1105920, 1119744, 1179648, 1209600, 1244160, 1290240, 1327104, 1382400, 1451520, 1474560, 1492992, 1572864, 1658880, 1679616, 1728000, 1769472, 1843200, 1866240, 1935360, 1966080, 1990656, 2073600, 2097152, 2177280, 2211840, 2239488, 2359296, 2419200, 2488320, 2580480, 2654208, 2764800, 2903040, 2949120, 2985984, 3110400, 3145728, 3317760, 3359232, 3456000, 3538944, 3628800, 3686400, 3732480, 3870720, 3932160, 3981312, 4147200, 4194304, 4354560, 4423680, 4478976, 4718592, 4838400, 4976640, 5160960, 5308416, 5529600, 5598720, 5806080, 5898240, 5971968, 6220800, 6291456, 6531840, 6635520, 6718464, 6912000, 7077888, 7257600, 7372800, 7464960, 7741440, 7864320, 7962624, 8294400, 8388608, 8709120, 8847360, 8957952, 9437184, 9676800, 9953280, 10077696, 10321920, 10368000, 10616832, 11059200, 11197440, 11612160, 11796480, 11943936, 12441600, 12582912, 13063680, 13271040, 13436928, 13824000, 14155776, 14515200, 14745600, 14929920, 15482880, 15728640, 15925248, 16588800, 16777216, 17418240, 17694720, 17915904, 18662400, 18874368, 19353600, 19906560, 20155392, 20643840, 20736000, 21233664, 21772800, 22118400, 22394880, 23224320, 23592960, 23887872, 24883200, 25165824, 25401600, 26127360, 26542080, 26873856, 27648000, 28311552, 29030400, 29491200, 29859840, 30965760, 31457280, 31850496, 33177600, 33554432, 33592320, 34836480, 35389440, 35831808, 37324800, 37748736, 38707200, 39191040, 39813120, 39916800, 40310784, 41287680, 41472000, 42467328, 43545600, 44236800, 44789760, 46448640, 47185920, 47775744, 49766400, 50331648, 50803200, 52254720, 53084160, 53747712, 55296000, 56623104, 58060800, 58982400, 59719680, 60466176, 61931520, 62208000, 62914560, 63700992, 66355200, 67108864, 67184640, 69672960, 70778880, 71663616, 72576000, 74649600, 75497472, 77414400, 78382080, 79626240, 79833600, 80621568, 82575360, 82944000, 84934656, 87091200, 88473600, 89579520, 92897280, 94371840, 95551488, 99532800, 100663296, 101606400, 104509440, 106168320, 107495424, 110592000, 111974400, 113246208, 116121600, 117964800, 119439360, 120932352, 123863040, 124416000, 125829120, 127401984, 130636800, 132710400, 134217728, 134369280, 139345920, 141557760, 143327232, 145152000, 149299200, 150994944, 152409600, 154828800, 156764160, 159252480, 159667200, 161243136, 165150720, 165888000, 169869312, 174182400, 176947200, 179159040, 185794560, 188743680, 191102976, 199065600, 201326592, 201553920, 203212800, 207360000, 209018880, 212336640, 214990848, 221184000, 223948800, 226492416, 232243200, 235146240, 235929600, 238878720, 239500800, 241864704, 247726080, 248832000, 251658240, 254803968, 261273600, 265420800, 268435456, 268738560, 278691840, 283115520, 286654464, 290304000, 298598400, 301989888, 304819200, 309657600, 313528320, 318504960, 319334400, 322486272, 330301440, 331776000, 339738624, 348364800, 353894400, 358318080, 362797056, 371589120, 373248000, 377487360, 382205952, 398131200, 402653184, 403107840, 406425600, 414720000, 418037760, 424673280, 429981696, 435456000, 442368000, 447897600, 452984832, 464486400, 470292480, 471859200, 477757440, 479001600, 483729408, 495452160, 497664000, 503316480, 509607936, 522547200, 530841600, 536870912, 537477120, 557383680, 566231040, 573308928, 580608000, 597196800, 603979776, 609638400, 619315200, 627056640, 637009920, 638668800, 644972544, 660602880, 663552000, 671846400, 679477248, 696729600, 707788800, ...

A000108: Catalan numbers

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324, 4861946401452, 18367353072152, 69533550916004, 263747951750360, 1002242216651368, 3814986502092304, 14544636039226909, 55534064877048198, 212336130412243110, 812944042149730764, 3116285494907301262, 11959798385860453492, 45950804324621742364, 176733862787006701400, 680425371729975800390, 2622127042276492108820, 10113918591637898134020, 39044429911904443959240, 150853479205085351660700, 583300119592996693088040, 2257117854077248073253720, 8740328711533173390046320, 33868773757191046886429490, 131327898242169365477991900, 509552245179617138054608572, 1978261657756160653623774456, 7684785670514316385230816156, 29869166945772625950142417512, 116157871455782434250553845880, 451959718027953471447609509424, 1759414616608818870992479875972, 6852456927844873497549658464312, 26700952856774851904245220912664, 104088460289122304033498318812080, 405944995127576985730643443367112, 1583850964596120042686772779038896, 6182127958584855650487080847216336, 24139737743045626825711458546273312, ...

A000110: Bell numbers

1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, 4506715738447323, 44152005855084346, 445958869294805289, 4638590332229999353, ...


References

[1] Neil J. A. Sloane, A Handbook of Integer Sequences, Academic Press (1973), ISBN 0-12-648550-X.

This book encouraged my developing interest in integer sequences, something that was already a hobby at age 9 after beginning to memorize the powers of 2, 3, 5, 6 and 7. It established many of the guidelines I still follow in my catalogs of sequences (notably MCS and nu-sequences), showing how to put sequences in a definitive order and other important ideas.

[2] Neil J.A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press (1995), ISBN 0-12-558630-2.



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