# Notable Properties of Specific Numbers

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(Meller's "370 property")

370 = (037 + 073 + 307 + 370 + 703 + 730)/6, which is the average of all possible permutations of its digits. It and several other multiples of 37 have this property for reasons that are discussed under that number. A larger example with a cool digit pattern is 456790123. Numbers with this property were first pointed out to me by Claudio Meller. Many larger examples are discussed here.

371 = 7×53 = 7 + 11 + ... + 53. Like 39, it is the product
of two primes p_{1} and p_{2} and the sum of all primes from
p_{1} to p_{2} inclusive. The next such number is
454539357304421 (see that entry for more).

371 is also the sum of the cubes of its own digits:
3^{3}+7^{3}+1^{3} = 371; see 153 for more like this.

This is a palindromic prime, the sum of the first three 3-digit palindromic primes, and a Woodall number. The good folks at Numberphile made a video, "383 is cool",

This is π^{4} + π^{5}, notable for being very close to e^{6}
(which is 403.428793...). This was first reported to
sci.math by Doug Ingram, who in July 1989 found it "[in someone's].sig file", most probably Soren G. Frederiksen of Ohio StateUniversity:

From the sci.math discussion thread it later entered theconsciousness of David Wilson, who (around 1997) reported it to Eric
Weisstein [167], from whom it made its way into
MathWorld^{119}, which describes the relation as

e^{6} - π^{4} - π^{5} = 0.000017673...

crediting Wilson. Another early reference is ^{52}.

This near-equality can be easily discovered with my RIES "equation finder" program, using the command:

ries 3.141592653589 --min-match-distance 1e-8 -NSCT

which tells it to find near (but not exact) equations for π without using the trig functions. Among its output are the equations:

x√π+1 = e^{3}/π

and

x^{2}/e^{3} = 1/√1+pi,

Since we're trying to approximate π, we can change each x to π
(and change the equal sign to a near-equality); then these equations
are both equivalent to π^{4}+π^{5} ≈ e^{6}. Another
equation output by the same RIES command is e^{x}-π =
4×5, which yields another curious near-integer.

See also 23.140692632779269005... and the Ramanujan constant.

This is close to π^{4} + π^{5}; see 403.428775....

A recurring "random number" in the webcomic Homestuck (the longest series in the MS Paint Adventures by Andrew Hussie). This page on the Homestuck wiki summarises appearances of the number.

For a while this number had a popular culture association with marijuana. Some anecdotes are at Urban Dictionary. See also 187.

This is a Catalan number, and shares with 27 the property that it is formed from the digits "4, 29" and is also the sum of the numbers from 4 through 29: 4+5+6+...+28+29 = (29-4+1)×(4+29)/2 = 26×33/2.

As reader Matt Goers points out, there aren't many such numbers: 15, 27, 429, 1353, 1863, 3388, ... (A186074 in Sloane's database) with about an average of three for each number of digits. 15 and 1353 fit a pattern which extends indefinitely: 133533 = 133+134+135+...+532+533 = (533-133+1)×(533+133)/2 = 401×666/2, and similarly for larger numbers like 13335333. If the first and last numbers are put together the other way (for example, 204 = 4+5+6+...+19+20 = (20-4+1)×(20+4)/2 = 204) we get another sequence of numbers (Sloane's A186076).

The smallest number that requires more than three terms to express
as a sum of 3-smooth numbers, as in X = 2^{a}3^{b} +
2^{c}3^{d} + 2^{e}3^{f} + ... (the next record-setters are 18431,
3448733, and 1441896119). This type of representation of a number is
referred to as a "double-base number system" (DBNS); see
here.

432 = 2^{4} 3^{3} which makes it 3-smooth. Like its
factors 108 and 216, it occurs occasionally in
religious and spiritual contexts, most often multiplied by some power
of 10 (see 43200, 432000,
4320000, and 4320000000). 108 doubles to 216
and then to 432; if you keep doubling further, you get 864,
1728, and 3456.

(an order-4 Kaprekar number)

An example of an order-4 Kaprekar number: take an
n-digit number, raise it to the 4^{th} power, divide the result into
4 groups of n digits each, add together and get the original number.
In the specific case of 433, the 4^{th} power is 35152125121; divide
this into groups of 3 digits (because 433 has 3 digits) and add:
35+152+125+121 = 433. A table of larger ones is located
here. See also 7776.

(the meta-Fibonacci triangle)

464 is the sum of row 7 of the following triangle, which is similar to the fibonomial triangle which in turn resembles Pascal's triangle. I refer to the numbers in this triangle as meta-fibonomials:

1 1 1 1 2 1 1 1 3 1 1 1 1 4 1 2 2 2 1 8 1 5 10 10 5 1 32 1 21 105 210 105 21 1 464 1 233 4893 24465 24465 4893 233 1 59184
These numbers are defined similarly to the fibonomials, but involve
terms of the form F_{Fn}, where F_{n} represents the n^{th}
Fibonacci number. For example, the 4^{th} element in the
7^{th} row (210) is
F_{F6}F_{F5}F_{F4}/F_{F3}F_{F2}F_{F1}
= F_{8}F_{5}F_{3}/F_{2}F_{1}F_{1} = 21×5×3/1×1×1.
The general form is
F_{Fn}...F_{Fn-k+1}/F_{Fk}...F_{F1}. This
formula always gives an integer, for reasons explained in the
fibonomial description.

The second number on each row is the sequence of F_{Fn}
(Sloane's A7570):
1,1,1,2,5,21,233,10946,5702887,.... The third number on each row
is F_{Fn}×F_{Fn-1}, a sequence that starts: 1, 2,
10, 105, 4893, 2550418, 62423801102, ... The 4^{th} number in each
row is F_{Fn}×F_{Fn+1}×F_{Fn+2}/2, a
sequence that starts 1, 2, 10, 210, 24465, 53558778, ....

If you start with a 3-digit number (with only a few exceptions) and
repeat the "Kaprekar transformation", you'll always end up with 0 or
with 495. In the Kaprekar transformation^{113}, you reorder the
digits largest-to-smallest, then subtract its reversal. For example,
starting with 143, the reordering is 431 and the reversal of this is
134; subtracting gives 431-134 = 297. Continuing in a similar manner:

431 - 134 = 297

972 - 279 = 693

963 - 369 = 594

954 - 459 = 495

954 - 459 = 495 (etc.)

This works for all 3-digit numbers except multiples of 111, provided that you treat an answer "99" as if it is a 3-digit number "099".

For 4-digit numbers the ending value is 6174. So far as I know, there are no unique ending values for numbers with fewer than 3 or with more than 4 digits. For example, numbers with 5 digits have no ending value (all starting values end in a cycle of 2 or more values); numbers with 6 digits have two possible ending values (549945 or 631764), though most end in cycles; etc.

495, 6174, 549945, 631764, ... are members of OEIS sequence A099009.

(From Matthew Goers)

(a perfect number)

The third perfect number, defined as a number that is equal to the sum of its proper divisors (divisors smaller than itself): 1+2+4+8+16+31+62+124+248=496. The search for perfect numbers was considered an important problem to the mathematicians of the classical Greek era. If the sum is less than the number, it is called deficient, and if the sum is greater, it is called abundant.

As you can see, in this case of 496, all of the divisors are either
powers of 2 (the 1, 2, 4, 8 and 16) or 31=2^{5}-1 times one of those
powers of 2. Euclid, working ca. 300 BC, found this pattern and showed
that if P is a prime number and if 2^{P-1} is also prime,
then X given by

X = 2^{P-1} (2^{P} - 1)

= 2^{N} (2^{N+1} - 1)

is a perfect number. More recently is was shown that all even perfect numbers fit this pattern.

However, not all numbers that fit the pattern are perfect; see 2047 and 8384512. It is still not known if there are any odd perfect numbers.

Prime numbers of the form 2^{P}-1 are called
Mersenne primes. See my
largenum notes for a (fairly) complete list of
known perfect numbers.

There are also numbers whose proper divisors add up to exactly twice the number (see 120) or three or more times (see 30240 and 154345556085770649600).

See also 6, 28, 8589869056, and the largest known perfect number.

This is 2^{9} and is also the sum of 7^{3} and
13^{2}. See also 728 and 4782969.

(a Carmichael number)

The first Carmichael number. Carmichael numbers are odd
composite numbers which cannot be found to be composite
using Fermat's Little Theorem, which states that for all
prime numbers p, if n is relatively prime to p, then
n^{p-1}-1 is divisible by p (sometimes stated: n^{p} - n
is divisible by p, which is an equivalent statement). For a
Carmichael number, no matter what n you pick this test will show
that it might be prime. See also 1729.

The largest known Wilson prime, a prime number p for which
(p-1)! + 1 is a multiple of p^{2}. As shown by
Wilson's theorem, all primes have the property that (p-1)! + 1
is divisible evenly by p, but Wilson primes have the additional
property that you can divide by p a second time. Other Wilson primes
include 5 and 13. The video by Numberphile
(5 13 and 563) points out that despite the belief
(based on statistical likelihood) that there "ought to be" an infinite
number of them, no other Wilson prime is known; all primes less than
2×10^{13} have been checked.

This number is the goroawase equivalent (by a mixture of kunyomi and onyomi readings) of Konami, the name of a videogame company: 5 = こ (ko), 7 = な (na), 3 = み (mi). The number appears in cheat codes for some Konami games on mobile phone platforms, and in Dance Dance Revolution (see Konami Code).

See also 9001, 73735963, and 1597463007.

576 = 4^{20}×3^{21}×2^{22}. It is also a member of
Somos' sequence A52129 associated with the constant
1.66168.... See also 1658880.

Leyland Primes

593 is a Leyland number because it can be expressed in
the form x^{y} + y^{x} (specifically, with x=9 and y=2 or
vice-versa). It is also prime, which makes it a "Leyland prime",
so-called because primes of this form have been
extensively studied
by British mathematician Paul Leyland. Numberphile has a
video about Leyland numbers here.

See also 32993, 5.19344×10^{15070} and
7.00558×10^{25049}.

(Torah by one gematria system)

Common letter-number value of the word Torah in Hebrew. There are
several Hebrew alphabetic number assignments used for Gematria
(numerology) but only one system for the common
purpose of communicating a number in ordinary text. It dates back to
before they used separate symbols for the digits the way they do now.
The assignment is as follows^{15}:

א=1 ב=2 ג=3 ד=4 ה=5 ו=6 ז=7 ח=8 ט=9 י=10 כ=20 ל=30 מ=40 נ=50 ס=60 ע=70 פ=80 צ=90 ק=100 ר=200 ש=300 ת=400

Using that assignment, the value of תורה (Torah) is 611 (400+6+200+5). See also 187, 216, and 304805.

Appears instead of 666 in some early Greek manuscripts of the relevant passage of the Bible. For more, see the 666 page and in particular its gematria section.

The first factor of a non-prime Fermat number. Fermat
conjectured that the Fermat numbers were all prime, and could
not factor 2^{32}+1=4294967297. In fact 4294967297
is composite, equal to 641 × 6700417. Euler showed that all factors of a
Fermat number 2^{2n}+1 must be of the form K2^{n+1}+1. In
this case, n is 5 and the factor 641 is equal to 10×2^{5+1}+1. For
more about this see the Wiki page on Fermat primes.

648 is the smallest number that can be expressed as a b^{a} in
two different ways: 3×6^{3} = 2×18^{2}. Such numbers are not
particularly common: 648, 2048, 4608, 5184, 41472, 52488, 472392,
500000, 524288, 2654208, 3125000, 4718592, ...

648 = 3×6^{3} = 2×18^{2}

2048 = 8×2^{8} = 2×32^{2}

4608 = 9×2^{9} = 2×48^{2}

5184 = 4×6^{4} = 3×12^{3}

41472 = 3×24^{3} = 2×144^{2}

52488 = 8×3^{8} = 2×162^{2}

472392 = 3×54^{3} = 2×486^{2}

500000 = 5×10^{5} = 2×500^{2}

524288 = 8×4^{8} = 2×512^{2}

2654208 = 3×96^{3} = 2×1152^{2}

3125000 = 8×5^{8} = 2×1250^{2}

4718592 = 18×2^{18} = 2×1536^{2}

10125000 = 3×150^{3} = 2×2250^{2}

13436928 = 8×6^{8} = 2×2592^{2}

21233664 = 4×48^{4} = 3×192^{3}

30233088 = 3×216^{3} = 2×3888^{2}

46118408 = 8×7^{8} = 2×4802^{2}

76236552 = 3×294^{3} = 2×6174^{2}

134217728 = 8×8^{8} = 2×8192^{2}

169869312 = 3×384^{3} = 2×9216^{2}

344373768 = 8×9^{8} = 3×486^{3} = 2×13122^{2}

402653184 = 24×2^{24} = 3×512^{3}

512000000 = 5×40^{5} = 2×16000^{2}

648000000 = 3×600^{3} = 2×18000^{2}

737894528 = 7×14^{7} = 2×19208^{2}

800000000 = 8×10^{8} = 2×20000^{2}

838860800 = 25×2^{25} = 2×20480^{2}

922640625 = 5×45^{5} = 3×675^{3}

1147971528 = 3×726^{3} = 2×23958^{2}

1207959552 = 9×8^{9} = 2×24576^{2}

1714871048 = 8×11^{8} = 2×29282^{2}

1934917632 = 3×864^{3} = 2×31104^{2}

2754990144 = 4×162^{4} = 3×972^{3}

3127772232 = 3×1014^{3} = 2×39546^{2}

3439853568 = 8×12^{8} = 2×41472^{2}

4879139328 = 3×1176^{3} = 2×49392^{2}

6525845768 = 8×13^{8} = 2×57122^{2}

6973568802 = 18×3^{18} = 2×59049^{2}

7381125000 = 3×1350^{3} = 2×60750^{2}

See also 344373768.

The square root of e to the power of 13. See 666
and 91. (From Raphie Frank^{98})

Main article: The number 666

Here are a few of the purely mathematical properties of 666:

666 is the sum of the squares of the first 7
primes: 2^{2} + 3^{2} + 5^{2} + 7^{2} + 11^{2} + 13^{2} +
17^{2}.

666 = 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + 6^{3} + 5^{3} + 4^{3}
+ 3^{3} + 2^{3} + 1^{3}. Such sums could be called "hyper-octahedral"
numbers, based on a 4-dimensional polyhedron analogous to the
octahedron.

It is the 36th triangular number: 1+2+3+...+36 = 666, which seems more significant because 36=6×6. 666 is the largest triangular number that is a repdigit. Because Roulette wheels have the numbers 1 through 36 (plus 0 and possibly also 00), 666 is the sum of the numbers on the Roulette wheel (and on a Roulette table, but not counting combination bets like "2nd 12" or "2-to-1").

See also 1000000000000066600000000000001 and
10^{(6.65565×10668)}

This is 26^{2}, the first square that is a
palindrome but whose square root is not a palindrome.

In the English language there are 26 letters, so 676 is the number
of combinations of 2 letters when order is distinct. If the standard
license plates in your area had 2 letters followed by 4 numbers, there
would be 26^{2}×10^{4} = 676×10000 = 6760000 possible plates.

"persistence" of numbers in Base 10

In 1973 N.J.A. Sloane described the process of multiplying the digits of a number together, and continuing until you get a non-changing result:

6×7×9 = 378; 3×7×8 = 168; 1×6×8 = 48; 4×8 = 32; 3×2 = 6; 6 = 6; 6 = 6; etc.

Starting with 679, we can perform the digit-multiplication five times until we get to a single-digit answer 6, and then the number no longer changes.

It's pretty easy to see that with every step the number gets smaller
until you're down to a single digit (since 9×9 is less than 100, any
three digits starting with 6 will give a product less than 600; a
similar argument shows that any N digits starting with A will give
a product less than A×10^{N-1}), and so any such sequence will
always end with a single-digit unchanging value.

Since it takes 5 steps starting with 679 to get to a single digit 6, Sloane says that the "persistence of 679 is 5. 679 is the smallest positive integer with a persistence of 5. The smallest number with a persistence of N for N=1, 2, 3, ... is OEIS sequence A003001: 10, 25, 39, 77, 679, 6788, 68889, ...

See also 39 and 277777788888899.

According to my classical sequence generator, 695 is the
next number after my childhood "favourite" numbers 7, 27
and 127. The formula it finds is: A_{0} = 0; A_{1} = -1;
A_{N+1} = N A_{N} - A_{N} + 2A_{N-1} + N (sequence
MCS27694341). This serves as an example of how easy
it is to find a sequence formula to match an arbitrary set of numbers.
See also 715 and 1011.

The first 7 prime numbers (2, 3, 5, 7, 11, 13, 17) can be arranged to form the factors of 714 (2×3×7×17) and 715 (5×11×13). Because of this, the primorial 2×3×5×7×11×13×17=510510 is also an oblong number : 714×715=510510. The repeated digits come from 1001 being a factor.

714=21×34, so it is a golden rectangle number.

In base 714 there are easy tests for divisibility by 9 different primes (the 7 listed above plus 23 and 31 because 23×31=713). See the 14, 21, 29 and 66 entries for more about these properties.

714 and 715 also have the property that their prime factors add up to the same total: 2 + 3 + 7 + 17 = 5 + 11 + 13. Another (smaller) example is 77 and 78: 77=7×11, 78=2×3×13, and 7+11=2+3+13. When one of the pair is divisible by a square, it matters how you count the factors. For example, 24 and 25 are a pair if you count each prime only once: 24=2×2×2×3, 25=5×5, and 2+3=5. 15 and 16 qualify if you count the multiples: 15=3×5, 16=2×2×2×2, and 3+5=2+2+2+2. But the pair 714 and 715 qualify regardless of which way you define it.

Such pairs are called Ruth-Aaron pairs because of Babe Ruth's
famous record of 714 career home runs, which was broken in 1974 when
Hank Aaron hit his 715^{th}. (Aaron reached 755 before retiring, but
the number 715 is almost as commonly associated with him; in 2007 his
record was surpassed by Bonds). Numberphile has a video on
the topic, Aaron Numbers featuring Carl Pomerance
who originally associated the names "Ruth-Aaron" to these pairs of
numbers.

In Sloane's database of integer sequences, the relevant entries are A6145, A39752, and A39753. See also 7129199.

Part of a Ruth-Aaron pair, see 714.

Follows 3, 7, 27 and 127 in sequence MCS55651588, which is an alternative solution to the "problem" described in the entry for 695 found via my classical sequence generator. Another alternative solution is A136580, which would give 747. (As with 695, these numbers are interesting just by being an example of the ease of finding formulas that match a mystery integer sequence. The choice of this example is just as arbitrary as for 695; see its description. For me, 3 was not a favourite number in childhood, except through its connection to 27.)

Like 120 and 210, 720 can be expressed as a product of consecutive integers in two distinct ways: 2×3×4×5×6 = 8×9×10. You can also add a 7 to both sides and get a similar equation whose value is 5040. Here are the smallest numbers with this property: 120, 210, 720, 5040, 175560, 17297280, 19958400, 259459200, 20274183401472000, 25852016738884976640000, 368406749739154248105984000000, ... This is OEIS sequence A64224. The sequence is infinite — see 19958400 for details as to why. I also have a page dedicated to these numbers.

This is one less than 729 which is 9^{3}, and thus it is
similar to 1729:

6^{3} + 8^{3} = 9^{3} + -1^{3}.

This example mysteriously appears in Ramanujan's lost notebook, on page 82 (see this article) despite that it does not fit the pattern of the formulas and the other examples on that page. See also 336365328016955757248.

As it is also 12^{3} - 10^{3}, 728 is expressible as a sum or
difference of two cubes in three ways:

728 = 6^{3} + 8^{3} = 9^{3} - 1^{3} = 12^{3} - 10^{3}

As the smallest positive integer with this property, 728 is the 3^{rd}
Cabtaxi number.

This is 3 to the power of 3!, a member of the fast-growing sequence
N^{N!} (OEIS sequence A53986; see also 281474976710656).

It is one of the cubes summing to Hardy's taxi number 1729, and the square of 27.

744 is related to the integers approximated by expressions of the
form e^{π√n} for certain special n, including most
famously 163. See 640320 and
Ramanujan constant.

One of the better-known numbers that is also a product or brand name: a well-known Boeing aircraft. 747 is also the sum of the first few even factorials: 0!+2!+4!+6! = 1+2+24+720 = 747. See also A136580 and 715.

757 is the smallest factor of 999999999999999999999999999=10^{27}-1
that is not also a factor of a smaller string of 9's, and therefore
757 is the smallest number whose reciprocal has a 27-digit
repeating decimal: 1/757=.00132100396301188903566710700132... This
is part of a series: 1/3 has a 1-digit pattern, 1/27 has a
3-digit pattern, and 1/757 has a 27-digit pattern. The series keeps
going up (because if 1/n has a d-digit pattern, n is always
larger than d), but computing the next number in the series is hard
because it is equivalent to factoring a large number like 10^{757}-1.
See also 7, 27 and 239.

The number of distinct, legal positions in Tic-tac-toe, ignoring rotations and reflections.

See also 26830.

Like 6, 12, 24, 48, 96, 192 and 384, 768 is a 3-smooth
number of the form 3×2^{n}. It is one of several such numbers to
occur in personal computer display dimensions (in this case, the
standard 1024×768 "XGA" mode); see also 192.

See 82944.

840 is a record-setter for having more divisors than
any smaller number, and it is the first such record-setter that does
not also appear as the number of divisors of another
record-setter^{99}. See also 12, 45360,
720720, 3603600, 245044800,
293318625600, 195643523275200,
278914005382139703576000,
2054221614063184107682218077003539824552559296000 and
457936×10^{917}.

841 = 29^{2} is the sum of two consecutive squares:
20^{2}+21^{2}. This is the second smallest example; the smallest is
5^{2} = 25 = 3^{2}+4^{2}. The series continues: 5^{2}, 29^{2},
169^{2}, 985^{2}, 5741^{2}, 33461^{2}, 195025^{2}, ... (Sloane's
integer sequence A1653; my
MCS13882118). This sequence consists of every other
Pell number; Each of these is 6 times the previous one minus
the one before that, for example, 169 = 6×29-5. See also 99
and 204.

854 is 5^{3}+9^{3}, and 855 is 7^{3}+8^{3}. Like 10744,
the two sums fit the form (2t^{2}-t-1)^{3}+(2t^{2}+t-1)^{3}
and (2t^{2}-1)^{3}+(2t^{2})^{3}; in this case t=2. (from
Vladimir Shevelev) See also 18426689288.

According to xkcd's Randall Munroe[216], "Jenny's Constant" is:

Munroe, for whom π is a favourite element of humorous formulas (for
example see xkcd 687), might have found
this formula using RIES. Putting 867.5309 into RIES directly
does not give this formula or anything like it, but 867.5309 divided
by π^{2} is 87.8992576343025..., and when this number is given to
RIES, it finds:

The phone number in the song is more commonly thought of as being the integer 8675309 (a twin prime, as Munroe handily notes in the mouseover text for the comic), but Munroe's formula is arguably better. Following the digits of the phone number itself, "Jenny's Constant" gives the year in which the song was recorded: 1981 (a fact noticed by Rob Johnson of the explainxkcd forums).

In an early published statement[134] of the "four 4's problem", it was stated that one could get as high as 112 using the standard rules (square roots allowed, concatenation e.g. two 4's make 44; decimal points e.g. three 4's make 4.4/.4 = 11); and if you allow subfactorials you can get as high as 877.

The smallest odd "abundant" number: the sum of its factors is larger than the number itself. The odd abundant numbers, Sloane's A5231, are: 945, 1575, 2205, 2835, 3465, 4095, ... There are far more even abundant numbers (Sloane's A5101).

952 is the sum of the cubes of its digits plus the product of its
digits: 952 = 9^{3}+5^{3}+2^{3}+9×5×2. (Thanks to
Cyril Soler for this tip)

The number of cells in the nematode worm C. elegans, heavily studied and used as a test animal in science. Barring rare mutation or harsh environmental conditions, every worm is exactly the same. It has 95 muscles and 302 neurons. The neurons signal the muscles at 1410 points and connect with each other at 6393 synapses. Its genome has been completely sequenced, with approximately 20405 protein-coding genes and precisely 100267633 DNA base pairs.

1/998 = 0.001002004008016032064128256513026052104208416833667334669..., a repeating decimal in which the powers of 2 appear one after another (until they start to overlap and break the pattern)

0.001 0.000002 0.000000004 0.000000000008 0.000000000000016 0.000000000000000032 0.000000000000000000064 0.000000000000000000000128 0.000000000000000000000000256 0.000000000000000000000000000512 0.000000000000000000000000000001024 0.000000000000000000000000000000002048 + ... ... --------------------------------------------- 0.001002004008016032064128256513026052...The same thing happens to a lesser degree in the digits of 1/98, and to a greater degree in the digits of 1/9998, 1/99998, etc. The reason for the pattern is easy to see if you consider how long division is performed, or just notice that 1/998 = (0.998+0.002)/998 = 0.001 + 2 × (0.001 × 1/998). The same phenomenon is responsible for the powers of 3 in the digits of 1/997, and so on. A similar pattern involving the Fibonacci numbers appears in the reciprocals of 89 and 9899. For more of this type of decimal fraction, see my separate article Fractions with Special Digit Sequences; see also 99.9998, 199, 9801, and 997002999.

Because 1002004... is similar to 1002001 which is the square of

1001, the square roots of numbers like 2/49.9 (where 2 and 499 are the factors of 998) have really cool digit patterns:

√2/49.9 = 0.2 002 003 005 008...

These digits are related to the series expansion of √1/(1-2x), as is explained in the entry for the square root of 62. See also 99.9998, 999998.

The casting out nines principle works for 99, 999, 9999 and so on. For the same reason that "casting" gives us an easy way to test for divisibility by 3, it also allows us to test easily for divisibility by 11, since 11 is a factor of 99, and for divisibility by the factors of 999, which are 27 and 37.

Because 27×37=999, the reciprocals of 27 and 37 contain each other as digits:

1/27 = 0.037037037037037...

1/37 = 0.027027027027027...

The same relationship exists between any two numbers whose product is
10^{N}-1 for some N. After the initial few, which are just minor
variations on 1/3 = 0.33333... and 1/9 = 0.11111..., a lot of more
interesting examples are found:

3 and 3: 1/3 = 0.333333...

1 and 9: 1/9 = 0.111111... and 1/1 = 0.999999...

9 and 11: 1/11 = 0.0909090909... and 1/9 = 0.1111111111...

99 and 101: 1/101 = 0.0099009900990099...
and 1/99 = 0.0101010101010101...

33 and 303: 1/303 = 0.0033003300330033...
and 1/33 = 0.0303030303030303...

271 and 369: 1/369 = 0.0027100271002710027100271...
and 1/271 = 0.0036900369003690036900369...

123 and 813: 1/813 = 0.0012300123001230012300123...
and 1/123 = 0.0081300813008130081300813...

693 and 1443: 1/1443 = 0.000693000693000693000693000693000693...
and 1/693 = 0.001443001443001443001443001443001443...

819 and 1221: 1/1221 = 0.000819000819000819000819000819000819...
and 1/819 = 0.001221001221001221001221001221001221...

2151 and 4649: 1/4649 = 0.000215100021510002151000215100021510002151...
and 1/2151 = 0.000464900046490004649000464900046490004649...

7227 and 13837: 1/13837 = 0.0000722700007227000072270000722700007227...
and 1/7227 = 0.0001383700013837000138370001383700013837...

7373 and 13563: 1/13563 = 0.0000737300007373000073730000737300007373...
and 1/7373 = 0.0001356300013563000135630001356300013563...

and so on. See also 99.9998 and 101.

999 is an example of a "repdigit", a number consisting of the same digit repeated a number of times. Repdigits often come up in the study of numbers that have properties relating to the sum of their digits, repeating decimal fractions, etc. For example, many Kaprekar numbers are repdigits, or are closely related to them; repdigits and factors thereof also appear in the sequences A7138 and A61075 related to repeating decimal fractions whose repeating part has N digits.

(a thousand)

Thousand is another number that, for most of us, temporarily holds the honor of being the biggest number we've heard of. Usually it replaces 100 in this role and is overtaken by 1000000.

When numbers get big, we tend to group the digits in 3's to make the
numbers easier to read: 134,217,728 instead of 134217728.
That causes people to pay a little more attention to groups of 3
digits than they otherwise would. For example, it's probably the main
reason why I discovered this formula for 2^{27}.
Sometimes it's almost as if we're working in base 1000.

The neighboring numbers 999 and 1001 also frequently play a role — if the phenomenon being investigated involves adding groups of 3 digits together, then it is related to divisibility by 999. If it involves a similarity between adjacent groups of 3 digits (such as 720720) it is related to divisibility by 1001. The two sometimes get inter-related when division is involved because 1/999 is 0.001001001001... and 1/1001 is 0.000999000999...; see also 999999.

1001 is a product of three consecutive primes: 1001 = 7×11×13. This sequence runs: 30, 105, 385, 1001, 2431, 4199, 7429, 12673, 20677, 33263, 47027, ... (OEIS sequence A46301). See also 77.

1001 is related in various ways to many numbers and number phenomena that have repeated sets of 3 digits, and some of these, like 720720, occur because 1001 is a product of three consecutive primes.

In One Thousand and One Nights, also known as 1001 Arabian Nights, concerns a very long period of time during which a series of stories and stories-within-stories are told by the narrator to avoid being killed by a king. There are many versions, including some that are actually organised into 1001 parts, all of suitable length for reading at bedtime and most of them ending in cliffhangers. 1001 nights is 143 weeks, nearly 3 years.

There is evidence of early versions of the Arabian Nights story collection, apparently an anthology of material originating in oral tradition, with far fewer than 1001 parts. 1001 was acquired and stuck because it represents "a lot" in a special way. The reader is probably familiar with similar uses of "101", "201", "501", and similar numbers in other contexts.

See also 999, 1000, 2001, 999999, 3603600, and this description of highly composite numbers.

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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.

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