Robert Munafo's Recent Additions to
Sloane's Database of Integer Sequences
This is a a small auxiliary database that my local search tools use
when I do lookups in the On-Line Encyclopedia of Integer Sequences.
This file was begun in December 2009 to fill my needs during the
period (Dec 2009 through roughly the beginning of Feb 2010) during
which the online server has been "frozen", pending the transition of
its services to the new OEIS wiki at http://oeis.org/wiki
I have many web pages related to sequences; including:
( www.mrob.com/pub/math/ then )
nu-sequences.html A listing of all the sequences I have worked on
seq-linrec2.html 2nd-order linear recurrence sequences
MCS.html A broad and complete listing of recurrence sequences ranked
by formula complexity
seq-coprime.html Mutually coprime sequences
seq-cullen.html Generalized Cullen and Woddall numbers
seq-wondrous.html Related to Collatz 3X+1 iteration
seq-narayana.html The Narayana triangle and several related triangles
seq-kaprekar.html Kaprekar sequences
seq-mandelbrot.html A few special sequences related to the Mandelbrot Set (more
at mrob.com/pub/muency/enumerationoffeatures.html)
seq-accelerate.html My efforts to make sequences that grow at quicker and quicker
rates as you proceed
seq-floretion Sequences generated by "floretions", a sort of double-quaternion
iteration with many variations
Here are pages on specific sequences that are particularly notable (lots of
original work, pretty illustrations, etc.)
seq-a000215.html Fermat numbers
seq-a001181.html Baxter permutations
seq-a002061.html Hogben's Centered Polygonal Numbers
seq-a005646.html Classifications of N Elements
seq-a006542.html C(n,3)C(n-1,3)/4
seq-a019296.html exp(pi*sqrt(N)) is nearly an integer
seq-a019473.html Still-Lifes with N cells in Conway's game of Life
seq-a020916.html "Molecules" -- a restricted class of permutations.
seq-a023394.html Prime Factors of Fermat Numbers
seq-a045619.html Product of 2 or more consecutive integers
seq-a052154.html Coefficients of Lemniscates for Mandelbrot Set, or Binary
Trees of Limited Height
seq-a064224.html Two Distinct Representations as Product of Consecutive Integers>1
seq-a082897.html Perfect Totient Numbers
seq-a092188.html Smallest Positive Integer M such that 2^3^4^5^...^N == M mod N
seq-a094358.html 2^^N == 1 mod N, or Squarefree products of factors of Fermat numbers
seq-a094534.html Centered Hexamorphic, or Automorphic Hexagonal, Numbers
seq-a100140.html Largest Denominator of Greedy Egyptian Fraction for M/N
seq-a160818.html Equal to Average of Permutations of its Digits
seq-a162002.html 2^^N == 2^(2^N) modulo N
The following also contain many illustrations and background material
related to sequences:
numbers.html Numbers (note in particular the entries for the numbers:
14, 21, 27, 29, 61, 66, 158, 2.003..*10^19728)
largenum.html What happens when you try to beat the integers in a race
towards infinity
Maintained by: Robert P. Munafo ( mrob.com/main-contact.html )
home page: www.mrob.com
Please note, this data is not auto-generated but is intended to be
compatible with programs that read the OEIS "eisBTfry0000.txt" files.
I have added a date field beginning with "%d" which can be safely
ignored. It is used to distinguish multiple archived previous versions
of individual A-number records.
Any entry without a "%I" field is an augmentation of another OEIS entry,
and should be concatenated with that version the create the full version.
Use the interface at www.research.att.com/~njas/sequences to retrieve
the main part of the record.
If an entry here has a "%I" field and the online OEIS also has a sequence
with the same number, then a sequence has been submitted with that
A-number. If the sequences are different then perhaps my reserved A-numbers
got reused.
If a "%S" field appears, then it (together with any accompanying %T
and %U field) should be taken as being a replacement for the %S,%T,%U
of the official online version. The same applies to %V,%W and %X
fields, if any. One set can be replaced without affecting the other.
All other fields are treated as additions, unless the XMl tag appears,
in which case the following text on that line is to be located
and deleted if it is found in the online version of that OEIS record. In
string-matching for this purpose, variations in whitespace are allowed but
no other variations are allowed. If the doesn't match anything it
is silently ignored.
The entry with number A999999 is a place-holder for my indexing
software.
(start)
%d A000055 20091230.123930
%C A000055 Also counts classifications of K items that require exactly N-1 binary partitions; see Munafo link at A005646, also A171871 and A171872.
%C A000055 The 11 trees for N=7 are illustrated at the Munafo web link.
%C A000055 Link to A171871/A171872 conjectured by Robert Munafo, then proven by Andrew Weimholt and Franklin T. Adams-Watters on Dec 29 2009.
%H A000055 R. Munafo, Relation to Tree Graphs
%Y A000055 Related to A005646; see A171871 and A171872.
%d A000698 20090112.090030
%Y A000698 Cvitanovic et. al. paper relates this seq. to A005411-A005414
%d A002487 20130126.013930
%Y A002487 Cf. A212288, A212289.
%d A002489 20100125.1556
%S A002489 1,1,16,19683,4294967296,298023223876953125,10314424798490535546171949056,256923577521058878088611477224235621321607,
%T A002489 6277101735386680763835789423207666416102355444464034512896,196627050475552913618075908526912116283103450944214766927315415537966391196809,
%U A002489 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,1019799756996130681763726671436132304456781416468067415248292558306065071863627636642030949423377254718066066358518538286207211
%C A002489 The values of "googol" in base N: "10^100" in base 2 is 2^4=16; "10^100" in base 3 is 3^9=19683, etc.
%C A002489 This is N^^3 by the "lower-valued" (left-associative) definition of the hyper4 or tetration operator (see Munafo webpage).
%H A002489 R. Munafo, Hyper4 Iterated Exponential Function
%d A005411 20090112.090030
%Y A005411 Cvitanovic et. al. paper relates this seq. to A000698 and A005412-A005414
%d A005412 20090126.033530
%Y A005412 Cvitanovic et. al. paper relates this seq. to A000698 and A005411-A005414
%d A005413 20090112.090030
%Y A005413 Cvitanovic et. al. paper relates this seq. to A000698 and A005411-A005414
%d A005414 20090126.033630
%Y A005414 Cvitanovic et. al. paper relates this seq. to A000698 and A005411-A005413
%d A005646 20091227.130727
%S A005646 1,1,1,3,6,26,122,1015,11847,208914,5236991,184321511
%C A005646 Extensive explanation with illustrations on Munafo web page.
%C A005646 This sequence is the row sums of triangle A171871.
%H A005646 R. Munafo, Classifications of N Elements
%Y A005646 Cf. A000055, A171872, A171873.
%E A005646 1015 term first calculated by Andrew Weimholt, Dec 15 2009
%E A005646 11847 term first calculated by Andrew Weimholt, Dec 19 2009
%E A005646 208914 term first calculated by Robert Munafo, Dec 29 2009
%E A005646 5236990 term (erroneous) from Robert Munafo, Dec 30 2009
%E A005646 "5236990" corrected to 5236991 by Robert Munafo, Jan 01 2010
%E A005646 184321511 term first calculated by Robert Munafo, Jan 10 2010
%d A008279 20100107.171430
%C A008279 F. T. Adams-Watters suggests adding the 'core' keyword, Jan 07 2010
%d A008949 20100530.041430
%C A008949 Row n, partial sum r gives the number of vertices within distance r (measured along the edges) of an n-dimensional unit cube, (i.e. the number of vertices on the hypercube graph Q_n whose distance from a reference vertex is <= r)
%d A010097 20100319.024030
%D A010097 D. E. Knuth, ``Supernatural Numbers'', in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 310-325.
%D A010097 D. E. Knuth, Selected Papers on Fun and Games, to be published (later in 2010)
%H A010097 R. Munafo, Alternative Number Formats, section on "Lexicographic Strings"
%Y A010097 Knuth articles also give A000918 and A171885
%d A034189 20100102.0509
%Y A034189 Cf. A171872 and A171876.
%d A034190 20100102.0509
%Y A034190 Cf. A171872 and A171876.
%d A039754 20091231.141330
%C A039754 For N=1 through N=5, the first 2^(N-1) terms of row N are also found in triangle A171871, which is related to A005646. This was shown for all N by Andrew Weimholt, Dec 30 2009.
%Y A039754 Cf. A171871
%e A039754 1, 1, 1; 1, 1, 2, 1, 1; 1, 1, 3, 3, 6, 3, 1, 1; ...
%e A039754 1, 1, 1; 1, 1, 2, 1, 1; 1, 1, 3, 3, 6, 3, 3, 1, 1; ...
%d A066186 20120311.1758
%e A066186 a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
%E a(4)=20 because A000041(4)=5 and 4*5=20.
%d A074753 20100112.131930
%C A074753 "sigma(k)" is A000203(k)
%d A078333 20100112.151630
%H A078333 Robert Munafo, Notable Properties of Specific Numbers
%H A078333 Robert Munafo, Notable Properties of Specific Numbers (entry for the number 1.632526919438)
%d A085279 20120130
%N A085279 Expansion of (1-2x-2x^2)/((1-2x)(1-3x)).
%F A085279 G.f.: (1-2*x-2*x*x)/((1-2*x)*(1-3*x))
%t A085279 Series[(1-2*x-2*x*x)/((1-2*x)*(1-3*x)), {x,0,26}]
%o A085279 (Macsyma) taylor((1-2*x-2*x*x)/((1-2*x)*(1-3*x)),x,0,26);
%d A094358 20110518.1330
%N A094358 Numbers n such that 2^^n == 1 mod n, where 2^^x is A014221(x).
%N A094358 Squarefree products of factors of Fermat numbers (A023394).
%C A094358 Note that the two names reflect a conjecture that is not proven. When I submitted A094358 to NJAS in 2004, it was called "Numbers n such that 2^^n == 1 mod n, where 2^^x is A014221(x)." Then later NJAS changed the name to "Squarefree products of factors of Fermat numbers". Thus, if the conjecture proves to be false, a new seq. will have to be created for "2^^n == 1 mod n".
%C A094358 641 is first member not in sequences A001317, A004729, etc.
%C A094358 Conjectured (by Munafo, see link) to be the same as: numbers n such that 2^^n == 1 mod n, where 2^^n is A014221(n).
%e A094358 3 is a member because it is in A023394. 51 is a member because it is 3*17 and 17 is also in A023394. 153=3*3*17 is not a member because its factorization includes two 3's.
%e A094358 See the Munafo web link for examples of the (conjectured) 2^^n == 1 mod n property.
%H A094358 Robert Munafo, Sequence A094358, 2^^A(N) = 1 mod N
%d A097486 20100107.211830
%S A097486 3,33,315,3143,31417,314160,3141593,31415927,314159266,3141592655,31415926537,314159265359,3141592653591
%C A097486 A(n) is an approximation of Pi*10^n. If you substitute "1/K" in place of "0.1" in the algorithm, the resulting sequence will approximate Pi*K^n. If expressed in base K, the sequence terms will then have digits similar to the digits of Pi in base K.
%C A097486 Calculating this sequence is subject to roundoff errors. In PARI/GP and in C++ using a quad-precision library, the value of A(7) is 31415927, not 31415928 as was originally recorded in this entry. - Robert Munafo mrob27(at)gmail.com, Jan 07 2010
%C A097486 In the PARI/GP program below, if you change "z=0" to "z=c" and "2.0" to "4.0", you get a similar sequence and in addition, A(-1)=0, which is "more aesthetically correct" given the notion that this sequence approximates Pi*10^n. However, such a modified program is NOT equivalent for positive N, it gives A097486(8)=314159267.
%H A097486 R. Munafo, Seahorse Valley
%t A097486 $MinPrecision=128; Do[c=SetPrecision[.1^n*I-.75,128]; z=0; a=0; While[Abs[z]<2, z=z^2+c; a++]; Print[a], {n,0,8}]
%o A097486 (PARI/GP) A097486(n)=local(a,c,z);c=0.1^n*I-0.75;z=0;a=0;while(abs(z)<2.0,{z=z^2+c;a=a+1});a
%o A097486 (Magma) A097486:=function(n) c:=10^-n*Sqrt(-1)-3/4; z:=0; a:=0; while Modulus(z)lt 2 do z:=z^2+c; a+:=1; end while; return a; end function;
%E A097486 Correction and more terms from Robert Munafo, Jan 08 2010
%E A097486 Terms through a(9) verified in Magma by Jason Kimberly, and in Mathematica by Hans Havermann
%E A097486 Mathematica code from Hans Havermann pxp(at)rogers.com
%E A097486 Magma code from Jason Kimberly Jason.Kimberley(at)newcastle.edu.au
%d A099152 20100106.163500
%S A099152 1,1,3,7,23,83,405,2113,12657,82297,596483,4698655,40071743,
%T A099152 367854835,3622508685
%E A099152 A(14) and A(15) from Ivica Kolar telpro(at)kvid.hr, Jan 06 2010
%d A136580 20100222.0756
%S A136580 1,1,3,7,27,127,747,5167,41067,368047,3669867,40284847,482671467,6267305647,87660962667,1313941673647,
%T A136580 21010450850667,357001369769647,6423384156578667,122002101778601647,2439325392333218667,51212944273488041647,
%U A136580 1126440053169940898667,25903229683158464681647,621574841786409380258667,15537113273014144448681647
%C A136580 Another extension of 3,7,27,127 (compare to MCS53617393 and MCS14342519156)
%d A169938 20100728.2144
%H A169938 R. Munafo, A puzzle sequence
%H A169938 Les-Mathematiques.net, Le saviez-vous? (forum discussion)
%d A171871 20091230.120130
%I A171871
%S A171871 1,0,1,0,0,1,0,0,1,2,0,0,0,3,3,0,0,0,3,17,6,0,0,0,1,36,74,11,0,0,0,1,60,
%T A171871 573,358,23,0,0,0,0,56,2802,7311,1631,47,0,0,0,0,50,10087,107938,83170,
%U A171871 7563,106,0,0,0,0,27,26512,1186969,3121840,866657,34751,235,0,0,0,0,19,55088
%O A171871 0,10
%N A171871 Triangle read by rows: Distinct classifications of N elements containing exactly R binary partitions
%C A171871 Significance of triangle suggested by Franklin T. Adams-Watters on 19 Dec 2009
%C A171871 Row N has N terms in this sequence. The triangle starts:
%C A171871 1
%C A171871 0,1
%C A171871 0,0,1
%C A171871 0,0,1,2
%C A171871 0,0,0,3,3
%C A171871 0,0,0,0,3,17,6
%C A171871 0,0,0,0,1,36,74,11
%C A171871 Value is A000055(N) when R=N-1 (last term in each row). (Conjectured by R. Munafo Dec 28 2009, then proven by A. Weimholt and F. T. Adams-Watters on Dec 29 2009)
%C A171871 Value is 1 when N=2^R.
%C A171871 Value is 1 when N=(2^R)-1.
%C A171871 Value is R when R>2 and N=(2^R)-2.
%C A171871 Value is A034198(R) when R>2 and N=(2^R)-3.
%C A171871 Conjecture by R. Munafo: In general, in each column, the last 2^(R-1) values are the same as the first 2^(N-1) values from the corresponding row of A039754. - R. Munafo, (mrob(AT)mrob.com) Dec 30 2009.
%C A171871 Value is 0 for all (N,R) for which N is greater than 2^R.
%C A171871 Each term A(N,R) can be computed most efficiently by first enumerating all classifications in A(N-1,R) plus those in A(N-1,R-1), and then adding an additional type and/or partition to each.
%H A171871 R. Munafo, Classifications of N Elements
%Y A171871 Cf. Row sums are A005646.
%Y A171871 Cf. A039754.
%Y A171871 Cf. Column sums are A171832.
%Y A171871 Last term in each row is A000055(N).
%Y A171871 Same triangle read by columns is A171872
%A A171871 Robert Munafo, Dec 29 2009 (submitted 20100121, pending)
%d A171872 20091230.123230
%I A171872
%S A171872 1,0,1,0,1,1,0,2,3,3,1,1,0,3,17,36,60,56,50,27,19,6,4,1,1,0,
%T A171872 6,74,573,2802,10087,26512,55088,91984,130267,157662,168890,
%U A171872 158658,133576,98804,65664,38073,19963,9013,3779
%O A171872 0,8
%N A171872 Triangle read by columns: Distinct classifications of N elements containing exactly R binary partitions.
%C A171872 The bottom of each column is marked by a single 0 in this sequence. Value is 0 for all (N,R) for which N is greater than 2^R.
%C A171872 See note on efficient computation in A171871.
%H A171872 R. Munafo, Classifications of N Elements
%Y A171872 First term in each column is A000055(R+1).
%Y A171872 Column 4 shares terms with A034189; column 5 with A034190.
%A A171872 Robert Munafo, Dec 29 2009 (submitted 20100121, pending)
%d A171873 20091230.124230
%I A171873
%S A171873 1,1,2,10,280,1173468
%O A171873 0,3
%N A171873 Column sums of triangle A171871
%C A171873 Next term is known to be greater than 220146725295227, based on link between A171871 and A039754.
%H A171873 R. Munafo, Classifications of N Elements
%A A171873 Robert Munafo, Dec 30 2009 (submitted 20100121, pending)
%d A171874 20091230.134830
%I A171874
%S A171874 0,0,0,1,1,2,4,7,16,46,174,3311,268446771,401906756202069927727330981
%O A171874 0,6
%N A171874 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171874 First 5 terms are {0,0,0,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171874 R. Munafo, Accelerating Sequences
%A A171874 Robert Munafo, Dec 31 2009 (submitted 20100121, pending)
%d A171875 20100101.141230
%I A171875
%S A171875 0,0,1,3,17,74,358,1631,7563,34751,160807
%O A171875 2,4
%N A171875 Subdiagonal of triangle A171871: Classifications of N elements containing exactly N-2 binary partitions.
%H A171875 R. Munafo, Classifications of N Elements
%A A171875 Robert Munafo, Jan 01 2010 (submitted 20100121, pending)
%d A171876 20100101.175330
%I A171876
%S A171876 1,1,1,1,1,3,3,1,1,4,6,19,27,50,56,1,1,5,10,47,131,472,1326,3779,9013,
%T A171876 19963,38073,65664,98804,133576,158658,1,1,6,16,103,497,3253,19735,
%U A171876 120843,681474,3561696
%O A171876 0,6
%N A171876 Mutual solutions to two classification counting problems: binary block codes of wordlength J with N used words; and classifications of N elements by J partitions.
%C A171876 This connection was conjectured by Robert Munafo, then proven by Andrew Weimholt.
%C A171876 A(n) counts 2-colorings of a J-dimensional hypercube with N red vertices and 2^J-N black, each edge has at most one red vertex. (Andrew Weimholt, Dec 30 2009)
%C A171876 This sequence contains terms of A039754 that are found in A171871/A171872. They occur in blocks of length 2^(J-1) as shown here:
%C A171876 1
%C A171876 1,1
%C A171876 1,1,3,3
%C A171876 1,1,4,6,19,27,50,56
%C A171876 1,1,5,10,47,131,472,1326,3779,9013,19963,38073,65664,98804,133576,158658
%H A171876 Harald Fripertinger, Enumeration of block codes
%H A171876 R. Munafo, Classifications of N Elements
%Y A171876 Cf. A039754, A171872, A171871, A005646
%A A171876 Robert Munafo, Jan 01 2010 (submitted 20100121, pending)
%d A171877 20100115.003630
%I A171877
%S A171877 0,0,1,1,1,3,5,9,25,73,423,61297,3814697357801,38288777744833624093154249190851262684887027625
%O A171877 0,6
%N A171877 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171877 First 5 terms are {0,0,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171877 R. Munafo, Accelerating Sequences
%A A171877 Robert Munafo, Jan 15 2010 (submitted 20100121, pending)
%d A171878 20100115.003830
%I A171878
%S A171878 0,0,0,0,1,2,3,6,13,33,120,765,4831534,55040353993453427047,
%T A171878 410186270246002225336426103593500672000000000000055040353997149550557
%O A171878 0,6
%N A171878 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)^a(n-5)
%C A171878 First 5 terms are {0,0,0,0,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171878 R. Munafo, Accelerating Sequences
%A A171878 Robert Munafo, Jan 15 2010 (submitted 20100121, pending)
%d A171879 20100115.004130
%I A171879
%S A171879 0,0,1,1,1,1,3,5,9,25,73,313,3263,1502337,278472902914281,
%T A171879 11984387434132924341157279996736444304839056033321
%O A171879 0,7
%N A171879 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)*a(n-5)^a(n-6)
%C A171879 First 6 terms are {0,0,1,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171879 R. Munafo, Accelerating Sequences
%A A171879 Robert Munafo, Jan 15 2010 (submitted 20100121, pending)
%d A171880 20100115.004330
%I A171880
%S A171880 0,0,0,1,1,1,2,4,7,16,46,166,1014,47066,12348246366,66716521529543607970475115226
%O A171880 0,7
%N A171880 a(n) = a(n-1) + a(n-2)*a(n-3) + a(n-4)*a(n-5)^a(n-6)
%C A171880 First 6 terms are {0,0,0,1,1,1}; thereafter apply the recurrence. Note that 0^0=1.
%H A171880 R. Munafo, Accelerating Sequences
%A A171880 Robert Munafo, Jan 15 2010 (submitted 20100121, pending)
%d A171881 20100115.054030
%I A171881
%S A171881 0,1,1,2,1,1,3,4,1,1,4,27,16,1,1,5,256,19683,256,1,1,6,3125,4294967296,
%T A171881 7625597484987,65536,1,1,7,46656,298023223876953125,340282366920938463463374607431768211456,
%U A171881 443426488243037769948249630619149892803,4294967296,1,1,8,823543
%O A171881 0,4
%N A171881 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=1.
%C A171881 n^^k is defined the left-associative way: n^^2=n^n, n^^3=(n^n)^n=n^(n^2), n^^4=((n^n)^n)^n=n^(n^3), and in general n^^k=n^(n^(k-1)).
%C A171881 More terms on Munafo website.
%C A171881 Array begins:
%C A171881 0,1,1,1,1,1,...
%C A171881 1,1,1,1,1,1,...
%C A171881 2,4,16,256,65536,...
%C A171881 3,27,19683,...
%C A171881 4,256,4294967296,...
%C A171881 5,3125,...
%C A171881 6,46656,...
%H A171881 R. Munafo, Hyper4 Iterated Exponential Function
%Y A171881 Cf. A171882.
%A A171881 Robert Munafo, Jan 15 2010 (submitted 20100121, pending)
%d A171882 20100116.012230
%I A171882
%S A171882 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,
%T A171882 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084096
%O A171882 0,4
%N A171882 Square array, read by antidiagonals, where T(n,k)=n^^k for n>=0, k>=0.
%C A171882 n^^k defined the right-associative way: n^^2=n^n, n^^3=n^(n^n), n^^4=n^(n^(n^n)), etc.
%C A171882 n^^0=1 by convention, so that n^^(k+1) = n^(n^^k) for all k>=0.
%C A171882 More terms on Munafo website.
%C A171882 Array begins:
%C A171882 1,0,1,0,1,0,1,...
%C A171882 1,1,1,1,1,1,1,...
%C A171882 1,2,4,16,65536,...
%C A171882 1,3,27,7625597484987,...
%C A171882 1,4,256,...
%C A171882 1,5,3125,...
%C A171882 1,6,46656,...
%H A171882 R. Munafo, Hyper4 Iterated Exponential Function
%Y A171882 Cf. A171881.
%A A171882 Robert Munafo, Jan 16 2010 (submitted 20100121, pending)
%d A171883 20100227.191630
%I A171883
%S A171883 3,29,24391,14510715208481,3055388613462301256452407743005777548691
%T A171883 28523273576637848665919896495441825882152454136941318837931307249186500390142888912756816872323385929633845346711703957
%N A171883 Mills primes, starting with 3.
%C A171883 For the standard Mills primes sequence, A051254, one starts with 2, and each successive term a(n) is the smallest prime greater than a(n-1)^3. This sequence uses the same definition but starts with 3.
%o A171883 (MAXIMA) n:3 $ l:10^100 $ print(n) $ while (nMills primes, starting with 3
%Y A171883 Cf. A051254
%O A171883 0,1
%A A171883 Robert Munafo, Feb 27 2010
%I A171884
%S A171884 0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,64,40,65,
%T A171884 39,66,38,67,37,68,36,69,35,70,34,71,33,72,32,73,31,74,30,75,29,76,28,
%U A171884 77,27,78,26,79,133,188,132,189,131,190,130,191,129,192,128,193,127,194
%N A171884 Lexicographically first injective and unbounded sequence a(n) satisfying |a(n+1)-a(n)|=n for all n
%C A171884 The map n -> a(n) is an injective map onto the nonnegative integers, i.e. no two terms are identical. That means that "unbounded" is redundant, except that they are bounded below by -1, i.e. the terms are non-negative.
%C A171884 The number of terms in these groups appear to be A008776.
%H A171884 R. Munafo, Lexicographically first injective and unbounded sequence A(n) satisfying |A(n+1)-A(n)|=n for all n
%H A171884 R. Munafo, main-A171884.c(C source code to generate the sequence)
%H A171884 R. Munafo, Re: seqs whose |differences| are 1,2,3,4,... (on seqfan mailing list)
%H A171884 N.J.A. Sloane, Re: seqs whose |differences| are 1,2,3,4,... (first message of the thread)
%e A171884 We begin with 0, 0+1=1, 1+2=3. 3-3=0 cannot be the next term because 0 is already in the sequence so we go to 3+3=6. The next could be 6-4=2 or 6+4=10 but we choose 2 because it is smaller.
%Y A171884 Cf. A005646, which allows occasional duplicate values. Cf. A118201, in which every value of a(n) and of |a(n+1)-a(n)| occurs exactly once, but does not ensure that the latter is strictly increasing.
%O A171884 1,3
%A A171884 Robert Munafo, Mar 11 2010
%Y A171884 Cf. A005132, which does not avoid duplications
%Y A171884 Cf. A118201, which avoids duplications by reordering the difference-values.
%d A171885 20100319.010830
%I A171885
%S A171885 0,1,4,5,24,25,26,27,112,113,114,115,116,117,118,119,480,481,482,483,484,
%T A171885 485,486,487,488,489,490,491,492,493,494,495,1984,1985,1986,1987,1988,1989,1990,
%U A171885 1991,1992,1993,1994,1995,1996,1997,1998,1999,2000,2001,2002,2003,2004,2005,2006,2007,2008,2009,2010,2011,2012,2013,2014,2015,8064,8065,8066,8067
%N A171885 Representation of n in D. E. Knuth's second prefix-unambiguous, order-preserving binary string system
%C A171885 The first two terms are the strings "00" and "01"; all others are binary strings beginning with "1".
%C A171885 With the important exception of a(1)=1, when expressed in binary, any value not appearing in this sequence appears as an initial substring of later terms. For example, 6 (110) is an initial substring of 27 (11011).
%E A171885 The string representations start: 0="00"; 1="01"; 2="100"; 3="101"; 4="11000"; 5="11001"; 6="11010"; 7="11011"; 8="1110000"; 9="1110001"; and so on. See the references for longer lists and fuller explanation.
%D A171885 D. E. Knuth, ``Supernatural Numbers'', in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 310-325.
%D A171885 D. E. Knuth, Selected Papers on Fun and Games, to be published (later in 2010)
%H A171885 R. Munafo, Alternative Number Formats, section on "Lexicographic Strings"
%Y A171885 Knuth's first system gives A000918 (less its initial term)
%Y A171885 Knuth's third system is A010097, the Levenshtein codes.
%O A171885 0,3
%A A171885 Robert Munafo, Mar 18 2010
%d A171886 20100530.033756
%I A171886
%S A171886 0,1,2,3,5,6,7,9,10,14,15,17,20,21,27,28,29,31,35,36,44,45,49,54,55,
%T A171886 65,66,71,77,78,90,91,97,104,105,119,120,121,127,135,136,152,153,161,
%U A171886 170,171,189,190,199,209,210,230,231,241,252,253,275,276,279,287,299
%N A171886 Numbers n such that A008949(n) is a power of 2.
%C A171886 Partial sums of binomial coefficients were considered in section 2.2 of the 1964 paper by Leech. The presence of the number 279 corresponds to the existence of the Leech lattice.
%C A171886 In general, A000217(n+1)+i-1 is in this sequence IFF the first i items in row n of Pascal's triangle add up to a power of 2.
%C A171886 Almost all members of this sequence are "trivial" terms of four types: A000217(i); A000217(i)+1, A000217(i)+i, and A000217(2i+1)+i for all integers i. 279 is the sole non-trivial term.
%E A171886 17 is in the sequence because A008949(17)=16, which in turn is because the first 3 elements of row 5 of Pascal's triangle, 1+5+10, add up to 16.
%E A171886 279 is in the sequence because the first 4 elements of row 24 of Pascal's triangle add up to 2^11: 1+23+253+1771=2048.
%D A171886 John Leech, ``Some Sphere Packings in Higher Space'', Can. J. Math., 16 (1964), page 669.
%H A171886 John Leech, Some Sphere Packings in Higher Space (PDF available from the publisher).
%O A171886 0,3
%A A171886 Robert Munafo, May 30 2010
$d A173279 20100222.0721
%I A173279
%S A173279 1,1,2,1,6,1,24,2,1,120,6,1,720,24,2,1,5040,120,6,1,40320,720,24,2,1,
%T A173279 362880,5040,120,6,1,3628800,40320,720,24,2,1,39916800,362880,5040,120,
%U A173279 6,1,479001600,3628800,40320,720,24,2,1,6227020800,39916800,362880,5040
%E A173279 Correction (duplicate of 8th row removed) by Robert Munafo, Feb 22 2010.
%d A176780 20100728.2144
%H A176780 R. Munafo, A puzzle sequence
%H A176780 Les-Mathematiques.net, Le saviez-vous? (forum discussion)
%d A178449 20101223.1731
%S A178449 1,744,-196884,167975456,-180592706130,217940004309743,
%T A178449 -19517553165954887,74085136650518742,-131326907606533204
%N A178449 Conjectured expansion of exp(Pi sqrt(163)) in powers of t, where t = 1/(640320)^3.
%H A178449 Jim Cullen, An approximation of pi from Monster Group symmetries
%o A178449 /* GNU bc code, computes a(0) through a(7) */
%o A178449 define trunc(x) { auto sc,t; sc=scale; scale=0; t=x/1; scale=sc; return(t) }
%o A178449 scale = 200; pi = 4 * a(1); r = e(pi * sqrt(163)); s = 640320;
%o A178449 c0 = 1 + trunc(r - s^3);
%o A178449 c1 = -1 - trunc(((s^3 + c0) - r) * s^3);
%o A178449 c2 = 1 + trunc((r - (s^3 + c0 + c1/s^3)) * s^6);
%o A178449 c3 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6) - r) * s^9);
%o A178449 c4 = 1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9)) * s^12);
%o A178449 c5 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12) - r) * s^15);
%o A178449 c6 = 1 + trunc((r - (s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12 + c5/s^15)) * s^18);
%o A178449 c7 = -1 - trunc(((s^3 + c0 + c1/s^3 + c2/s^6 + c3/s^9 + c4/s^12 + c5/s^15 + c6/s^18) - r) * s^21);
%E A178449 Cullen link, bc code, and a(8) from Robert Munafo (mrob27(AT)gmail.com), Dec 23 2010
%d A181783 20100607.155830
%I A181783
%S A181783 1,3,1,2,11,3,5,1,10,1,2,5,3,5,1,2,4,1,1,2,4,51,1,4,2,2,31,1,3,1,1,5,
%T A181783 1,1,1,14,1,1,4,2,2,8,1,1,3,23,1,1,4,16,2,1,2,13,2,1,1,1,3,1,1,4,3,2,
%U A181783 1,1,36,1,2,1,1,1,2,3,2,1,3,3,31,2,1,2,2,2,2,1,2,1,1,1,5,1,6,6,1,2,2,
%N A181783 Continued fraction for sqrt(phi)
%O A181783 1,2
%A A181783 Robert Munafo, Jun 7 2010
%d A181784 20101221.180632
%I A181784
%S A181784 1,1,4,22,140,969,7084,53820,420732,3782992,32389076,275617830,
%T A181784 2350749914,20140518790,173429992350,1500850805160,14550277251918,
%U A181784 133009333771170,1198324107797254
%N A181784 Numerators of a series sum related to a game of chance.
%C A181784 Consider a 1-dimensional random walk from 0 with equal-probability steps of Pi and -1. One way to compute the probability of eventually walking below 0 is as the sum over n of the probabilities of becoming negative after a walk with exactly n steps of Pi (n >=0) and max(ceil(n*pi),1) steps of -1. The total number of walks of such length for a given n is 2^(n+max(ceil(n*pi),1)), or 2^(n+A004084(n)) (n>=1), forming a sequence of denominators, and this sequence gives the numerators, the number of possible sequences of length (n+max(ceil(n*pi),1)) drawn from {Pi, -1} such that no partial sum except the total sum is < 0.
%C A181784 See the Munafo web page for complete description.
%C A181784 a(n) diverges from A002293 because pi is not exactly 3.
%H A171884 R. Munafo, Related to a Game of Chance
%H "My Math Forum" discussion thread, I give, duz... what is it?
%H "duz" blog entry, Random Walking
%e Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...
%K nonn,frac
%A A181784 Robert Munafo, Dec 21 2010
%O A181784 0,3
%d A181785 20110508.002313
%I A181785
%S A181785 1,1,2,5,10,25,48,107,193,365,621,1082,1715
%N A181785 Wechsler's "convex-hull polyominoes": convex hull contains no additional grid points.
%N A181785 Wechsler's "disklike" polyominoes: convex hull contains no additional grid points.
%C A181785 Given a polyomino P on a square lattice, if you replace each of the squares in P with a point (say the "upper-left" corner) and call that set of points S, then define H to be the convex hull of S: the polyomino is said to be "disklike" if all lattice points in H are also in S.
%E A181785 For N=5 there are 12 polyominoes, but only 10 qualify. The two that do not are the U and V pentominoes, pictured here:
%E A181785 . * . * . . . * * *
%E A181785 . * * * . . . * . .
%E A181785 . . . . . . . * . .
%E A181785 Both are "concave" in the sense that a convex hull of the 5 points in the pentomino also includes one grid point that is not in the pentomino.
%H A181785 R. Munafo, Wechsler's Convex-Hull Polyominoes
%A A181785 Robert Munafo, May 7 2011
%E Initial entry by _Robert Munafo_, May 08 2011
%E Name changed (with Wechsler's approval) by _Robert Munafo_, May 12 2011
%E a(14)-a(16) added by _Robert Munafo_, May 12 2011
%O A181785 1,3
%d A182369 20120502.2227
%N A182369 Decimal expansion of (7^(e - 1/e) - 9)*Pi^2, also known as Jenny's constant.
%C A182369 The next digit is a 0, and the following 4 digits (1, 9, 8, 1) are the year the song was recorded (1981). (Noticed by Rob Johnson of the explainxkcd.com forums)
%C A182369 Randall Munroe almost certainly used my RIES program to find this one. He would have started with 8675309 and divided by pi, then by pi^2, and shift the decimal point to various positions and feed each number into RIES. Thus he would have tried 0.2761436620, 2.761436620, 27.61436620, etc. and then 0.8789925763, 8.789925763, 87.89925763, etc. This last one gives the formula 7^(e-1/e)-9 which he picked because of the symmetry you can get from "e/1-1/e". The match is close enough to get the desired 7 digits 867.5309, but it has to be complete coincidence that Randall Munroe found a formula that gives the phone number followed by the year the song was recorded.
%H A182369 Robert Munafo, 867.5309019816854...
%H A182369 explain xkcd, Approximations [Note: This site is a Wiki and as such, there is no author to this article, and the website specifically avoids giving specific ownership of articles to individual people. So, "explain xkcd" is the author, and it apparently is not capitalized.]
%d A211074 20130126.005649
%C A211074 Links and refs are in chronological order, because that's better.
%D A211074 D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486–492.
%H A211074 D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Aug 14 1999.
%H A211074 J. Cserti, Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors, first published Sep 8 1999.
%H A211074 Anonymous, Infinite 2D square grid of 1-ohm resistors, April 30 2004 (this date is on an old archive.org copy).
%H A211074 Michael Pusateri, Google Labs Aptitude Test, Sep 13 2004.
%H A211074 Alan Eustace, Pencils down, people (official Google Blog article, archived on archive.org), Sep 30 2004.
%H A211074 Peter G. Doyle and J. Laurie Snell, Random walks and electric networks (2006), section 1.1.4.
%H A211074 Randall Munroe, Nerd Sniping, Dec 12 2007.
%H A211074 Matthew Beckler, Infinite Grid of Resistors - Solution by Simulation, Dec 16 2009.
%H A211074 Kevin Brown, Infinite Grid of Resistors, date unknown.
%d A557274 20091231.114930
%S A557274 3,5,11,9,8,25,12,4,16,19,13,22,7,26,14,10,15,2,6,24,23,18,21,17,1,20
%N A557274 Most common to least common letters in the qwerty board (numbers labeled)
%C A557274 Alphabetic keys are numbered from the top-left: Q=1, W=2, E=3, R=4, etc. ... and A=11, S=12, etc.
%e A557274 A[1]=3 because "E" is the most commonly used letter, and it is the 3rd key (counting from the left on the top row) on a QWERTY keyboard.
%E A557274 Comment and example added by R. Munafo, Dec 31 2009.
%E A557274 Unusual sequence number pointed out to NJAS by R Munafo on Dec 31 2009.
%E A557274 Reassigned to A170835 on Jun 1 2010.
R800000 is a template for a new entry in the R8xxxxx section
%d R800000 20130205.123456
%I R800000
%S R800000 0,1
%N R800000 .
%A R800000 Robert Munafo, Mmm dd YYYY
Sequences from "R800001" through "R899999" are ones I plan to add soon, but
do not yet have official A-numbers.
%d R800001 20130205.123456
%I R800001
%S R800001 0, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
%N R800001 Numerators of Taylor series expansion of 2*x*e^x - 2*e^x + 2
%C R800001 This Taylor series is useful for deriving an initial approximation for the Lambert W function (see Veberic paper).
%D R800001 Darko Veberic, Having Fun with Lambert W(x) Function, arXiv:1003.1628v1 [cs.MS] 8 Mar 2010. [From _Robert Munafo_, Feb 5 2013]
%o R800001 (MAXIMA) w1(y) := y*%e^y $ f(y) := 2*(%e*w1(y-1)+1) $ t1 : taylor(f(y), y, 0, 20) $ makelist(ratnumer(coeff(t1,y,n)), n, 0, 20);
%A R800001 Robert Munafo, Mmm dd YYYY
%d R800002 20130205.123456
%I R800002
%S R800002 1, 1, 1, 3, 4, 15, 72, 420, 2880, 22680, 201600, 1995840, 21772800,
%T R800002 259459200, 3353011200, 46702656000, 697426329600, 11115232128000,
%U R800002 188305108992000, 3379030566912000, 64023737057280000
%N R800002 Denominators of Taylor series expansion of 2*x*e^x - 2*e^x + 2
%C R800002 This Taylor series is useful for deriving an initial approximation for the Lambert W function (see Veberic paper).
%D R800002 Darko Veberic, Having Fun with Lambert W(x) Function, arXiv:1003.1628v1 [cs.MS] 8 Mar 2010. [From _Robert Munafo_, Feb 5 2013]
%o R800002 (MAXIMA) w1(y) := y*%e^y $ f(y) := 2*(%e*w1(y-1)+1) $ t1 : taylor(f(y), y, 0, 20) $ makelist(ratdenom(coeff(t1,y,n)), n, 0, 20);
%A R800002 Robert Munafo, Mmm dd YYYY
%d R800003 20140412.132100
%I R800003
%S R800003 6, 26, 53, 214, 429, 1718, 6874, 13749
%N R800003 Decimal values of binary strings produced by Turing Google Doodle (rabbit sequence program)
%A R800003 Robert Munafo, Apr 12 2014
https://oeis.org/A272853 (formerly R800004)
%d A272853 20160508.014900
%I A272853
%N A272853 Ramanujan's alpha-series
%S A272853 9, 791, 65601, 5444135, 451797561, 37493753471, 3111529740489
%T A272853 258219474707159, 21429104870953665, 1778357484814447079
%U A272853 147582242134728153849, 12247547739697622322431
%O A272853 0,1
%C A272853 Generated by the same G.f. as A051028 (the a-series) but using the Laurent series, i.e. the series expansion has negative powers of x.
%C A272853 These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n)
%C A272853 They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).
%D A272853 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272853 Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
%F A272853 G.f.: f(x)=(1+53x+9x^2)/(1-82x-82x^2+x^3).
%F A272853 a(-3)=-11161; a(-2)=-135; a(-1)=-1; a(n)=82*a(n-1)+82*a(n-2)-a(n-3).
%E A272853 a(3)=5444135 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%o A272853 (Wolfram|Alpha) Series[(1+53*a+9*a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272853 Cf. A051028, A051029, A051030.
%Y A272853 Cf. A272854, A272855.
%A A272853 Robert Munafo, May 8 2016
https://oeis.org/A272854 (formerly R800005)
%d A272854 20160508.015000
%I A272854
%N A272854 Ramanujan's beta-series
%S A272854 10, 812, 67402, 5593538, 464196268, 38522696690, 3196919629018
%T A272854 265305806511788, 22017185020849402, 1827161050923988562
%U A272854 151632350041670201260, 12583657892407702716002
%O A272854 0,1
%C A272854 Generated by the same G.f. as A051030 (the c-series) but using the Laurent series, i.e. the series expansion has negative powers of x. It is mislabeled as "gamma" in Ramanujan's notes.
%C A272854 These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n)
%C A272854 They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).
%D A272854 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272854 Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
%F A272854 G.f.: f(x)=(2+8x-10x^2)/(1-82x-82x^2+x^3).
%F A272854 a(-3)=14258; a(-2)=172; a(-1)=2; a(n)=82*a(n-1)+82*a(n-2)-a(n-3).
%E A272854 a(3)=5593538 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%o A272854 (Wolfram|Alpha) Series[-1*(2+8a-10a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272854 Cf. A051028, A051029, A051030.
%Y A272854 Cf. A272853, A272855.
%A A272854 Robert Munafo, May 8 2016
https://oeis.org/A272855 (formerly R800006)
%d A272855 20160508.015100
%I A272855
%N A272855 Ramanujan's gamma-series
%S A272855 12, 1010, 83802, 6954572, 577145658, 47896135058, 3974802064140
%T A272855 329860675188578, 27374461238587818, 2271750422127600332
%U A272855 188527910575352239722, 15645544827332108296610
%O A272855 0,1
%C A272855 Generated by the same G.f. as A051029 (the b-series) but using the Laurent series, i.e. the series expansion has negative powers of x. It is mislabeled as "beta" in Ramanujan's notes.
%C A272855 These give identities of the form alpha(n)^3 + beta(n)^3 = gamma(n)^3 + (-1)^n, where alpha(n)=A272853(n), beta(n)=A272854(n) and gamma(n)=A272855(n)
%C A272855 They are from page 82 of the "lost notebook" of Ramanujan. A051028,A051029,A051030 give his examples (135, 138, 172) and (11161, 11468, 14258) while A272853,A272854,A272855 give the examples (9, 10, 12), (791, 812, 1010), and (65601, 67402, 83802).
%D A272855 S. Ramanujan, The Lost Notebook and Other Unpublished Papers (1988), p. 341. New Delhi (Narosa publ. house).
%H A272855 Robert Munafo, Sequences Related to the Work of Srinivasa Ramanujan
%F A272855 G.f.: f(x)=(2-26x-12x^2)/(1-82x-82x^2+x^3).
%F A272855 a(-3)=11468; a(-2)=138; a(-1)=2; a(n)=82*a(n-1)+82*a(n-2)-a(n-3).
%E A272855 a(3)=6954572 because 5444135^3 + 5593538^3 = 6954572^3 - 1.
%o A272854 (Wolfram|Alpha) Series[-1*(2-26a-12a^2)/(1-82*a-82*a^2+a^3), {a, Infinity, 10}]
%Y A272855 Cf. A051028, A051029, A051030.
%Y A272855 Cf. A272853, A272854.
%A A272855 Robert Munafo, May 8 2016
X900000 is a template for a new entry in this file
%d X900000 20130101.123456
%I X900000
%S X900000 0,1
%N X900000 .
%A X900000 Robert Munafo, Mmm dd YYYY
Sequences from "X900001" and up are just to hold information I want to
show up in my searches, but I am not planning to submit to OEIS. In
some cases these are valuable sequences, but I'm not submitting them
because I expect that they will be submitted by the people who did the
primary work on them.
%d X900001 20100105.211430
%I X900001
%S X900001 1,2,4,8,16,35,77,179,440,1160,3264,9950,33206,121943,494011,2235399,11391306,65287199,422908306,3130775625,26490210964,255257056748,2825013955541,36147331371446,531237157370531,8965348473026888,175629366371057918,3992619892094868709,104457049859201232450,3162313206707299567108
%N X900001 Graph Counting Problem
%C X900001 Described by Franklin T. A-W. on seqfan
%C X900001 "Consider graphs where each node is labeled with a positive integer; labels may be used more than once. Two nodes with the same label may not be connected by an edge. Graphs are distinguished up to permutations of nodes with the same label. How many graphs are there with labels totaling n?"
%o X900001 PARI-GP code from Max Alekseyev:
%o X900001 { part2freq(p) = local(r);
%o X900001 r = vector(sum(i=1,#p,p[i]));
%o X900001 for(j=1,#p, r[p[j]]++);
%o X900001 r
%o X900001 }
%o X900001 { a(n) = local(s,p,k,P,r);
%o X900001 s=0;
%o X900001 p=partitions(n);
%o X900001 for(i=1,#p,
%o X900001 k=part2freq(p[i]);
%o X900001 k = select(x->(x>0),k);
%o X900001 P = vector(#k,j,partitions(k[j]));
%o X900001 for(j=1,#k, for(t=1,#P[j], P[j][t] = part2freq(P[j][t]) ));
%o X900001 forvec(v = vector(#k,j,[1,#P[j]]), \\ fix cycle structures
%o X900001 r = 1 / prod(j=1,#k, prod(t=1,#P[j][v[j]], t^P[j][v[j]][t] * %o X900001 P[j][v[j]][t]! ));
%o X900001 for(j=1,#k, for(t=j+1,#k,
%o X900001 r *= prod(ii=1,k[j], prod(jj=1,k[t], 2^(gcd(ii,jj) * %o X900001 P[j][v[j]][ii] * P[t][v[t]][jj]) ));
%o X900001 ));
%o X900001 s += r;
%o X900001 );
%o X900001 );
%o X900001 s
%o X900001 }
%E X900001 Terms 1160, 3264 and 9950 from "Hugo" hv(at)crypt.org
%E X900001 Terms beyond 9950 are from Max Alekseyev
%A X900001 Author Name, Mmm dd YYYY
%d X900002 20100105.211930
%I X900002
%S X900002 1,2,3,6,12,28,65,173,496,1527,5092
%N X900002 Connected graphs satisfying criteria for X900001
%C X900002 See X900001 for more
%A X900002 Author Name, Mmm dd YYYY
%d X900003 20100107.171528
%I X900003
%S X900003 1,8,6,10,14,12,4,20,16,24,18,22,28,26,34,30,32,36,40,42,46,38,44,52,48,54,50,58,56,62,64,60,66,68,72,70,74,80,76,78,86,82,84,90,92,94,88,98,96,104,
%T X900003 100,102,108,110,112,114,106,116,122,118,120,124,126,130,132,134,128,138,136,142,140,144,146,148,150,154,152,156,158,160,162,164,168,166,172,174,170,176,182,180,178,184,188,192,186,194,190,196,198,
%U X900003 202,204,200,208,206,210,212,214,216,218,220,222,224,226,230,232,228,234,236,238,242,240,244,246,248,252,254,250,256,258,262,264,260,266,270,268,272,274,280,276,282,284,278,286,290,292,288,294,298,300
%O X900003 1,2
%N X900003 The lexicographically first sequence of natural numbers such that no set of consecutive sequence members adds to a prime, and no number occurs in the sequence more than once.
%C X900003 Idea by Eric Angelini in seqfan; correction and more terms by Robert Munafo
%C X900003 Franklin T Adams-Watters suggests the definition "a(1) = 1; a(n) is the smallest even number not yet used such that the cumulative sums are all non-prime"
%Y X900003 cf. X900004, X900005, X900006
%E X900003 First 50 terms from Robert Munafo
%E X900003 First 1000 terms from Charles Greathouse
%A X900003 Eric Angelini Eric.Angelini(at)kntv.be, Jan 07 2010
%d X900004 20100107.171528
%I X900004
%S X900004 1,8,6,6,4,8,6,6,4,6,8,6,6,6,4,6,4,4,6,6,4,4,4,6,4,8,4,8,6,6,4,6,8,4,8,6,4,4,4,4,4,4,6,4,8,4,6,6,6,8
%O X900004 1,2
%N X900004 The lexicographically first sequence of natural numbers such that no set of consecutive sequence members adds to a prime.
%C X900004 Based on original statement of problem by Eric Angelini in seqfan (which did not state "no number occurs in the sequence more than once" but gave the first few terms making it clear that X900003 is what was intended)
%Y X900004 cf. X900003, X900005, X900006
%A X900004 Eric Angelini (with Robert Munafo), Jan 07 2010
%d X900005 20100107.171528
%I X900005
%S X900005 1,8,6,9,4,10,12,14,16,15,20,18,21,22,24,25,27,26,28,30,32,34,33,35,36,38,39,40,42,45,44,46,48,49,51,50,52,54,55,56,57,58,60,62,63,65,64,66,68,70,69,72,75,74,76,77,78,80,81,82,84,85,86,87,88,90,91,92,
%T X900005 93,94,95,96,98,99,100,102,104,105,106,108,110,111,112,114,116,115,117,118,119,120,121,122,123,124,125,126,128,129,130,132,133,134,135,136,140,138,141,142,143,144,145,146,147,148,152,150,153,154,155,
%U X900005 156,158,159,160,162,161,164,165,166,168,169,171,170,172,174,175,176,177,178,180,182,183,184,185,186,187,188,189,190,192,194,195,196,198,200,201,202,203,204,205,206,207,209,208,210,212,214,213,215,216
%N X900005 Variation of X900003: cumulative sums and terms must be non-prime, but other subsequences have no such restriction.
%C X900005 Suggested by Franklin T. Adams-Watters in response to discussion of X900003
%A X900005 Franklin T. Adams-Watters franktaw(at)netscape.net, Jan 07 2010
%d X900006 20100107.171528
%I X900006
%S X900006 1,3,2,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,
%T X900006 70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,
%U X900006 127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175
%N X900006 Variation of X900003: "a(1) = 1; a(n) is the smallest even number not yet used such that the cumulative sums are all non-prime"
%C X900006 Identical to A000027 after first two terms; cumulative sums are 1 and 4 together with the triangular numbers A000217 from 6 onward.
%C X900006 Suggested by Franklin T. Adams-Watters
%A X900006 Franklin T. Adams-Watters franktaw(at)netscape.net, Jan 07 2010
%d X900007 20100107.171528
%I X900007
%S X900007 1,2,3,5,7,4,13,17,9,8,19,11,15,14,21,23,25,10,31,35,27,16,29,37,45,20,47,39,33,12,43,49,53,6,41,51,59,22,61,55,73,34,65,57,67,38,71,63,69,24,77,81,75,18,83,87,89,48,93,101,95,28,107,117,79,44,91,97,
%T X900007 85,40,109,103,115,26,105,113,123,32,99,119,129,36,125,111,137,46,127,121,139,52,131,135,143,30,141,147,145,54,133,155
%N X900007 Smallest positive integer not already in sequence with a(n)+a(n+1)+a(n+2)+a(n+3) prime
%E X900007 More terms from Franklin T. Adams-Watters franktaw(at)netscape.net, Jan 07 2010
%Y X900007 Cf. A055265
%A X900007 Eric Angelini Eric.Angelini(at)kntv.be, Jan 07 2010
%d X900008 20100120.015630
%I X900008
%S X900008 1,2,4,8,16,28,49,84,144,252,441,777,1369,2405,4225,7410,12996,22800,
%T X900008 40000,70200,123201,216216,379456,665896,1168561,2050657,3598609,
%U X900008 6315113,11082241,19448018,34128964
%N X900008 Number of subsets S of {1,2,3,...,n} with the property that if x is a member of S then at least one of x-2 and x+2 is also a member of S
%C X900008 By considering n-bit bit-strings that avoid 010 and that do not start with 10 or end with 01, it can be shown that a(2*n)=A005251(n+2)^2 and a(2*n+1)=A005251(n+2)*A005251(n+3) - Andrew Weimholt andrew.weimholt(at)gmail.com, Jan 19 2010
%A X900008 John W. Layman layman(at)math.vt.edu, Jan 19 2010
%d X900009 20100311.053930
%I X900009
%N X900009 Temporary name of A171884
%d X900010 20130127.215033
%I X900010
%S X900010 345, 3436, 23432, 2, 456548, 234, 7
%N X900010 Buy spammo(TM) brand genuine imitation pills!
%A X900010 Charles Greathouse, Mmm dd YYYY
%d X900011 20130127.215346
%I X900011
%S X900011 193, 344, 111, 992, 232, 159, 160, 756, 349
%N X900011 27 random digits I made up in a drowsy stupor, then capriciously severed into groups of three.
%A X900011 Robert Munafo, 2013 Jan 24
%d X900012 20130131.020000
%I X900012
%S X900012 10, 14, 15, 17, 18, 30, 34, 38, 40, 43, 45, 50, 51, 53, 73, 135
%T X900012 137, 149, 180, 304, 314, 317, 319, 335, 337, 338, 339, 345, 350
%U X900012 370, 380, 400, 434, 445, 504, 505, 508, 509, 514, 515, 517, 518
%N X900012 Numbers which when displayed on a calculator and viewed upside-down resemble words from the National Scrabble Association Dictionary
%H X900012 Mike Wolfberg, All 101 Two-Letter Words (with brief definitions)
%A X900012 Robert Munafo, 2013 Jan 30
%d X900013 20130101.030000
%I X900013
%S X900013 6, 15, 6, 6, 12, 15, 18, 27, 12, 18, 6
%N X900013 Sum of digits of the LCM of the Nth Pythagorean triple
%C X900013 Pythagorean triples are from A103606; thus the LCM of (3,4,5) is 60, and sum of digits is 6; the LCM of (5,12,13) is 780 and sum of digits is 15, etc.
%A X900013 Robert Munafo, Jan 31 2013
Use X900000 as a template for a new X90xxxx sequence
%d A999999 99999999.999999
%I A999999
%S A999999 0,1
%N A999999 .
(end)
Robert Munafo's Recent Additions to
Sloane's Database of Integer Sequences
See the large block of explanatory text at the top of this file.
(xref: ~/shared/proj/OEIS/0-overview.txt )