# Notable Properties of Specific Numbers

First page . . . Back to page 9 . . . Forward to page 11 . . . Last page (page 25)

144 is 12^{2}, and also happens to be the 12^{th}
Fibonacci number by the standard definition. See also
61917364224.

One of the largest numbers, that is not a power of 10, that has a specific word (gross) assigned to it. Moser is another.

Gratuitous connection to 27: 144 = (3+3+3+3)×(3+3+3+3); rearrange the parentheses to get 3+3+3+(3×3)+3+3+3 = 27.

145 = 1!+4!+5!; see 40585.

149 is prime, and is also prime when reversed (941). Such numbers are called Emirps. A palindrome prime such as 101 does not count.

150 = 2×3×5^{2}. It has no prime factors larger than 5,
and this makes it a 5-smooth number. 5-smooth numbers include the
3-smooth numbers plus: 5, 10, 15, 20, 25, 30, 40, 45, 50,
60, 75, 80, 90, 100, 120, 125, 135, 150, 160, 180, 200, 225, 240, 250,
270, 300, 320, 360, 375, 400, 405, 450, 480, 500, 540, 600, 625, 640,
675, 720, 750, 800, 810, 900, 960, ... That list is Sloane's
A80193 and is generated by multiplying all the
3-smooth numbers by 5, 25, and other higher powers of 5. The complete
list of 5-smooth numbers is Sloane's A51037.

153 = 5!+4!+3!+2!+1!.

153 = 1^{3}+5^{3}+3^{3}. It is the smallest number that is the sum
of powers of its own digits, where the power is the same as the number
of digits, aside from the trivial 1-digit cases like 8=8^{1}. The next
few numbers with this property are: 370, 371, 407, 1634, 4150, 4151,
8208, 9474, 54748, ... (Sloane's A23052). See also
4679307774 and
115132219018763992565095597973971522401.

153 = 17+16+15+...+3+2+1, the 17^{th} triangular
number. It is also 100+28+25, and appears in a New Testament story as
a number of fish. A lot has been made of this in connection with the
Enneagram, a system of personality type classification.

Gratuitous connections to 27: 15^{3}/(1×5^{3}) = 27;
or 15^{3}/(1×5×3) = 225 and 22+5 = 27.

158 is the sum of row 7 of the "fibonomial triangle". This is a triangle of numbers similar to Pascal's triangle, using Fibonacci factorials in place of normal factorials. The numbers in the triangle are called fibonomials:

1 sum: 1 1 1 sum: 2 1 1 1 sum: 3 1 2 2 1 sum: 6 1 3 6 3 1 sum: 14 1 5 15 15 5 1 sum: 42 1 8 40 60 40 8 1 sum: 158 1 13 104 260 260 104 13 1 sum: 756 1 21 273 1092 1820 1092 273 21 1 sum: 4594 1 34 714 4641 12376 12376 4641 714 34 1 sum: 35532
These numbers are defined similarly to the binomial coefficients in
Pascal's triangle but using the
Fibonacci numbers F_{1}=1, F_{2}=1, F_{3}=2,
F_{4}=3, etc. (more here). For example, the
4^{th} element in the 8^{th} row (260) is
F_{7}F_{6}F_{5}/F_{3}F_{2}F_{1} = 13×8×5/2×1×1.
The general form is F_{n}...F_{n-k+1}/F_{k}...F_{1}. It is
not immediately obvious that this formula always gives an integer. It
does — because of the property of fibonacci numbers
that if A is divisible by B, Fv∀ is divisible by F_{B},
combined with the fact that the ordinary binomial coefficients use the
formula {n}...(n-k+1)/k...1, which itself is always an integer
for somewhat simpler reasons.

On Pascal's triangle, the second number on each row is the sequence
of integers; here they are the Fibonacci number. On Pascal's triangle
the following number is an oblong number; here the 3^{rd}
items (1, 2, 6, 15, 40, 104, 273, ...) are
golden rectangle numbers. The next numbers after that
(1, 3, 15, 60, 260, 1092, ...) are
F_{n}×F_{n+1}×F_{n+2}/2.^{68},^{69}

The same transformation can be done again to create the meta-fibonomial triangle.

163 appears in the "Ramanujan constant"
e^{pi×√163}, which is very nearly an integer. It
is the largest Heegner number, a set of 9 integers that share this
same property. See e^{π}, and
262537412640768743.999999... for a description of this
and some related amazing numbers.

The appearance of √163 in the Ramanujan constant is related to √-163 and the Monster group, due to the Monstrous moonshine phenomena.

√-163 can also be used to demonstrate that x^{2}-x+41
yields prime numbers for all x from 1 to 40.^{95}

163 = 1 + 2 × 3 ^{4} can be produced on most "algebraic"
calculators by keying 1 + 2 × 3 x^{y} 4 =. If you have an RPN
calculator you would key in either 1 2 3 4 x^{y} × + or
4 3 2 1 + × x^{y} (depending on how the x^{y} key works).
See also 10^{1.0979×1019}.

Gratuitous connection to 27: 163_{10} = 61_{27} and 61_{10} =
27_{27} — or another way of saying the same thing: 2×27+7 = 61,
and 6×27+1 = 163

168 is the order of the group PSL(2,7), which describes the symmetries of the "Fano plane" and the automorphisms of the "Klein quartic". These are two important abstract geometric objects.

The Klein quartic is the solution to the equation
x^{3}y+y^{3}+x=0 with x,y complex, and its surface is like
a 3-toled torus (as opposed to a normal torus with just one hole). It
is a two-dimensional surface that can be covered with a finite number
of heptagons, joined 3 at each vertex — like a warped
Dodecahedron made from heptagons. It takes 24 heptagons to
do this[180], and you end up with 84 edges and
56 vertices. No other surface can be covered with a smaller
number of heptagons in a regular way. It has 168 automorphisms because
you can "rotate" the tiling around the center of any of the heptagons,
and there are 24 heptagons to choose from: 7×24=168. (On a
dodecahedron or most of the other "normal" Platonic solids, when you
rotate one face, one other face stays put and rotates the other way.
Paridoxically, when you rotate one of the heptagons in the Klein
quartic, two others stay put, with one rotating twice as far and the
other rotating four times as far (or three times as far in the
opposite direction)!)

The Fano plane is a set of seven lines and seven points
(another illustration is on John Baez' page [166])
with the property that any two lines uniquely determine a point and
any two points uniquely determine a line. It is useful (among other
things) as a way to show the symmetry of the basis elements of the
Octonions and how their multiplication rules work^{45}. Imagine
that the 7 points have been labeled 1 through 7 and the 7 lines have
been labeled A through G. When this is done, there are 21 unique
letter-number combinations like A1, A2, A4, B1, B3, etc.
have been chosen out of the complete set of 49. The Fano plane's
168-fold symmetry is evident in how many ways you can relabel the
points and lines such that the same 21 out of 49 letter-number groups
occur. It is easy to see that line A has to still have points 1, 2
and 4 on it; but you can pick any of the 7 lines to be line A. With
a little work you can see that if you stick to this choice of which
line is labeled A, you can pick from all 4×3×2×1=4!
combinations of where to place the labels 3, 5, 6 and 7. Thus there
are 7×24 equivalent labelings.

Of course, 168=24×7 is also the number of hours in a week. See 10080.

169 is a "powerful" number, meaning that all of its prime factors
occur at least twice. It's pretty easy to see that all squares are
powerful, in this case we have 13^{2}. Similarly 144 = 12^{2}
= 2^{4}×3^{2}. More generally, anything expressible as
X^{2}Y^{3}, where X and Y are positive integers, is a powrful
number. The powerful numbers, Sloane's A1694, include many
highly composite numbers.

173.297929 ≈ 29.530588853 × 223 / 38

(eclipse season interval in days)

The number of mean solar days between each time the nodes of the Moon's orbit align with the sun. This is the length of time between each eclipse season (the time of year when lunar and/or solar eclipses occur). The oldest known calendars of the Sumerians of Mesopotamia were based on the synodic month and the eclipse season (which was called by the same name that later became used for the solar year). Near the equator, the tropical year does not matter too much, but the moon was very important inasmuch as it provides light at night and eclipses were considered a very important event. The eclipse season is a little less than half a year, because the tilt of the moon's orbit keeps shifting; after 18 years it comes around to the same time of year again.

The magic constant of a 3×3 magic square that contains all prime numbers:

17 89 71 113 59 5 47 29 101When the primes in this square are listed in order (5, 17, 29, 47, 59, 71, 89, 101, 113) a simple pattern can be seen: The successive differences are 12, 12, 18, 12, 12, 18, 12, 12. The values 12 and 18 are simple multiples of the primorial 2×3=6, and the symmetry of their arrangement is also important.

See also 199.

Like 6, 12, 24, 48 and 96, 192 is a 3-smooth number of
the form 3×2^{n}, which causes it to be seen a little more often
than other numbers of its size. It is also the first of several such
numbers to occur in personal computer display dimensions (the Apple ][
hires graphics mode produced 192 rows of pixels); see also
768.

A member of the Lucas-Lehmer sequence defined by A_{0}=4 and
A_{n+1}=A_{n}^{2}-2. This sequence starts 4, 14, 194, 37634,
1416317954, 2005956546822746114, ... (Sloane's A3010). It is
used in a test to determine if a Mersenne number is
prime; that test is described here and some
examples are given here for examples. See also
47.

196 is a "4-dimensional pyramidal number" given by the sum:
6+5×2^{2}+4×3^{2}+3×4^{2}+2×5^{2}+6^{2} = 6 + 20 + 36 + 48 +
50 + 36. This is similar to the formula for the
tetrahedral numbers (for example,
6+5×2+4×3+3×4+2×5+6 = 56). It is because of this that squares
stacked in 4-dimensions in a manner similar to the diagram
here can be used to form a sequence of 4-dimensional
pyramids corresponding to these numbers. There is also a general
formula A_{n-1} = n^{2}(n^{2}-1)/12. The sequence starts: 1,
6, 20, 50, 105, 196, 336, 540, ... (Sloane's
A2415).

Take any number and reverse its digits, and add the two numbers
together. For example, starting with 129, 129 plus 921 is 1050.
Continue until you get a palindrome: a number which is equal to its
own reversal: 1050+0501 = 1551, so 129 takes two steps. Some numbers
take quite a while to arrive at a palindrome, and 196 is the first
that goes so far that it is unknown whether it ever arrives at a
palindrome. (In other bases it is possible to prove that certain
numbers go on forever. In base 2 the number 22_{10} = 10110_{2}
produces an infinite sequence.)

199 is prime, and if you add 210 over and over again, you get 8 more primes. These 9 primes together can be arranged into a 3×3 magic square, and is the smallest possible magic square made of primes in arithmetic progression:

1669 199 1249 619 1039 1459 829 1879 409See also 177.

199 is also a Woodall number: 2×10^{2}-1.

The reciprocal 1/199 exhibits a pattern similar to that of 1/998. Here are the digits, broken into groups to show the structure; the repeating pattern is 99 digits long:

1/199 = 0.005 025 125 628140703517587939698492462311557788944 7236180904522613065326633165829145728 64 32 16 08 04 02 01 005 025 125 6281407...

Since 1/199 = 5/995, we have the digits of 1/995 (which contain powers
of 5, for the reason explained under 998) multiplied by 5,
which just means that the 0^{th} power of 5 does not appear. The
powers of 2 also appear, in the opposite order at the end, for reasons
originating in the fact that 2×5=10 and we are working with base 10
digits.

204 = 17*12, the product of the numerator and denominator of 17/12, which is an approximation to the square root of 2 formed by the Pell numbers and another similar sequence. The first few numbers like this are: 1, 6, 35, 204, 1189, 6930, 40391, 235416, 1372105, 7997214, ... (Sloane's integer sequence A1109). The sequence follows the pattern:

A_{n} = 6 A_{n-1} - A_{n-2}

(for example, 204=6×35-6 and 1189=6×204-35).

A really nifty property of these numbers is that their squares are also triangular — see 41616 for more.

Approximate mass ratio between an electron an a muon. The muon is about 207 times heavier, as a result when it binds with a nucleus its charge distribution is about 207 smaller. This means that a muon bound with a proton forms an atom that is chemically similar to hydrogen but with 207 times smaller diameter. See also 137.03599....

210 is a primorial: the product of the first N primes: 210=2×3×5×7. The primorials are: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, ... (Sloane's A2110).

210 is a 3-d oblong number, the product 5×6×7. Curiously, it is also 1×2×3 + 2×3×4 + 3×4×5 + 4×5×6 (see 3024 for more).

Since it is also 14×15, it is another example of a number that is the product of consecutive integers in two different ways (see also 120, 720, 5040).

For a while, 2 is the most common gap between consecutive
prime numbers. Then (starting at about 100, and permanently
after 1000), gaps of 6 occur most often. Up at around
10^{35}, it shifts to 30. There is a conjecture that above
about 10^{425} the most common prime gap is 210.

Base 210 is also a record-setter for tests for divisibility by primes. See 66.

216 is a cube, 6^{3}, and is the sum of three consecutive
cubes^{72}: 3^{3} + 4^{3} + 5^{3} = 6^{3}. See also 25,
143, 8000 and 31858749840007945920321.

8 pieces are required to construct a puzzle that can be assembled as a 6×6×6 cube and reassembled into the three smaller cubes. The 6 "slices" of the puzzle would appear as follows, with letters a through h representing the eight puzzle pieces:

f f f f f h f f f f f h f f b b b b f f f f f e f f f f f e f f b b b b f f f f f e f f f f f e f f f b b b g g g e e e g g g e e e g g g b b b g g g e e e g g g e e e g g g e e e g g g e e e g g g e e e g g g e e e slice-1 slice-2 slice-3 f f b b b b a a b b b b a a b b b b f f b b b b a a b b b b a a b b b b c c b b b b c c b b b b c c b b b b c c b b b b c c b b b b c c b b b b c c c e e e c c c e e e c c c d d d c c c e e e c c c e e e c c c d d d slice-4 slice-5 slice-6
216 = 8×27 = 2^{3}×3^{3}, and had spiritual significance

in ancient times because it was the product of the first two cubes
(they didn't count 1^{3} as a cube).

It is also 6×6×6, and is the value of Φ(666), the number of integers less than and relatively prime to 666.

In the 1998 movie π (a.k.a. "Pi the Movie"), the Qabbalistic Jews are searching for a sequence of 216 Hebrew letters, or a 216-digit number (which is the same thing in ancient Hebrew, since the letters of the Hebrew alphabet are used to represent the digits of numbers).

220 and 284 form an amicable pair, an idea that goes back to the time of Pythagoras. If you add the factors of 220 you get 284 and vice versa. The search for amicable numbers is closely related to the classical question of finding perfect numbers.

231 appears as a denominator in the "greedy" Egyptian fraction
for 3/7: ^{3}/_{7} = ^{1}/_{3} + ^{1}/_{11} + ^{1}/_{231}.

To the best of our knowledge, the ancient Egyptians had no general
notation for an arbitrary fraction A/B. They had a special symbol for
^{2}/_{3}, and any other fractions were expressed as a sum of
"unit fractions", fractions of unity, such as ^{1}/_{3} and ^{1}/_{7}.
For example, instead of ^{3}/_{4} they wrote ^{1}/_{2}+^{1}/_{4}, and
instead of ^{5}/_{6} they wrote ^{1}/_{2}+^{1}/_{3}. By convention,
they always expressed it as a sum of different unit fractions (for
example, ^{2}/_{7} could not be written ^{1}/_{7}+^{1}/_{7}, but
^{1}/_{4}+^{1}/_{28} or ^{1}/_{5}+^{1}/_{20}+^{1}/_{28} were OK), and
the fractions were always listed largest-first (that is, smallest
denominators first).

It is not entirely known why they did it this way. It could have
been for practical reasons in performing physical divisions. For
example, imagine dividing 2 equal bushels of grain among 7 people.
Each person should get ^{2}/_{7} of a bushel. The simple approach
would be to divide each into 7 equal parts (give each person ^{1}/_{7}
plus ^{1}/_{7}). But a fair division into 7 parts is difficult; 4 is
much easier (use a balance, or the "I split, you choose" method, to
divide in half; then repeat). So, divide each bushel into 4 equal
parts, and give ^{1}/_{4} bushel to each person; the remaining
^{1}/_{4} bushel then is divided into 7, which is much easier than the
original task. ^{2}/_{7} = ^{1}/_{4} + ^{1}/_{28}.

It is possible to express any fraction as a sum of unit fractions,
and usually in more than one different way. There are several
different algorithms for converting a fraction like ^{3}/_{7} into
an Egyptian fraction sum. One, called the "greedy" algorithm, works by
subtracting the largest possible unit fraction each time and repeating:

Given: Fraction X/Y, where X and Y are positive integers and X

2: 1/A is the first (or next) term in the Egyptian fraction.

3: Compute X/Y - 1/A and reduce to lowest terms. The result is
the new X/Y. If nonzero, go back to step 1 and repeat.

This algorithm always produces an answer with a finite number of terms,
but the denominators sometimes get really large. Starting with ^{3}/_{7}
we get:

^{3}/_{7} = ^{1}/_{3} + ^{2}/_{21}

^{2}/_{21} = ^{1}/_{11} + ^{1}/_{231}

therefore ^{3}/_{7} = ^{1}/_{3} + ^{1}/_{11} + ^{1}/_{231}

233 is a Fibonacci number and is also prime.
The prime Fibonacci numbers are: F_{3}=2, F_{4}=3,
F_{5}=5, F_{7}=13, F_{11}=89, F_{13}=233,
F_{17}=1597, F_{23}=28657, F_{29}=514229,
F_{43}=433494437, ... (Sloane's A5478). You may notice that
(except for F_{4}=3), each prime Fibonacci number has a prime
index. This is related to a more general property of Fibonacci numbers
(see ^{20} and [138]):

for N≥3, F_{M} is divisible by F_{N}
if (and only if) M is divisible by N

This prime number appears in one of the earliest known geometrically converging formulas for computing π:

π/4 = 4 arctan(1/5) - arctan(1/239)
= SUM [ (-2^{n})/(2n+1)(5^{2n+1}) - (-1^{n})/(2n+1)(239^{2n+1}) ]

This formula works because of this special relationship between 5 and 239 through their squares:

2 × 13^{4} = 239^{2} + 1 = 57122

and

2 × 13 = 5^{2} + 1 = 26

and

arctan(x) + arctan(y) = arctan( (x + y) / (1 - x y) )

This relationship makes it possible to show that a geometric
construction of 4 right triangles in the proportion 1 :: 1/5 ::
(√26)/5 and one triangle in the proportion 1 :: 1/239 ::
(√57122)/239 can be used to produce a triangle in the proportion
1 :: 1 :: √2. Also related to this is the fact that 239/169 =
239/13^{2} is a good close approximation to the
square root of 2.

You might wonder, why do all that when you could compute π/4 directly from the arctangent of 1:

π/4 = arctan(1)
= SUM [ (-1^{n})/(2n+1) ]
= 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

The reason is that this series converges too slowly to be of any
use. The 5^{2n+1} in the denominator in the series above makes it
converge quickly enough so that you get about 1.398 = log_{10}(5^{2})
digits of π for each term you evaluate.

239 is the smallest factor of 9999999 that is not also a factor of some smaller string of 9's, and therefore 239 is the smallest number whose reciprocal has a 7-digit repeating pattern: 1/239=0.00418410041841004184... See also 27 and 757.

Here are two rather obscure properties of 239:

239 is the largest number that cannot be expressed as the sum of 8
(or fewer) cubes (it requires 9: 239 = 5^{3} + 3^{3} + 3^{3} + 3^{3} +
2^{3} + 2^{3} + 2^{3} + 2^{3} + 1^{3}). The only other such number is
23. 239 also requires the maximum number of terms to be expressed as a
sum of squares (4 squares, 239 = 15^{2} + 3^{2} + 2^{2} +
1^{2}) or of 4th powers (19 4th powers). It does not require the
maximum number of 5th powers.

Define the "sum of prime factors" sopf(N) to be the sum of each of the prime factors of N, counting a prime more than once if N is a multiple of its square, cube, etc. So, for example, sopf(42) = 2 + 3 + 7 = 12, and sopf(27) = 3 + 3 + 3 = 9. sopf(N) = N only if N is prime, or if N is 1 or 4. Now consider the sum N + sopf(N). As N increases, this value increases irregularly. Some values never occur — for example, there is no N such that N + sopf(N) equals 12 — and other values occur more often — for example, there are two 14's and two 23's. The value 239 occurs often enough that if you add the sopf(N)'s for all N's that have N+sopf(N)=239, you get a sum greater than 239. 239 is the first number for which this occurs (the next few are 1439, 2159, 4319).

(Personal: 239 is another street number where I have lived. The
famous HAKMEM^{72} file was MIT AI memo number 239.)

240 is a "Fibonacci factorial", the product of the first 6 Fibonacci numbers: 1×1×2×3×5×8 = 240. See also 158.

It is known that in 8-dimensional space, the greatest number of
spheres of equal size that can touch a single central sphere is 240.
The arrangement that produces this is called the
E_{8} lattice, a
repeating arrangement of points in 8-dimensional space that has the
symmetries of the E8 group, a mathematical object
that has applications in theoretical physics. See also 24 and
196560.

243 = 3^{5} = 9×27, and has a few cool properties.
It is a perfect totient number, along with all
other powers of 3. There is more about these numbers on a separate
page here.

If we divide 5^{3} by 3^{5}, we get a 27-digit repeating
decimal with the digits 514403292181069958847736625. These 27 digits
taken as an integer have the 370 property: the average of
all possible permutations of its digits is equal to itself.

First page . . . Back to page 9 . . . Forward to page 11 . . . Last page (page 25)

Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×10

^{11}10

^{18}5.4×10

^{27}10

^{40}5.21...×10

^{78}1.29...×10

^{865}10

^{40000}10

^{9152051}10

^{1036}10

^{1010100}— — footnotes Also, check out my large numbers and integer sequences pages.

s.11