| Notable Properties of Specific Numbers |
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There is only one possible magic hexagon (defined similarly to a
magic square:
3
17 19
18 7 16
1 2
11 5 12
6 4
9 8 10
14 13
15
When you add the 3, 4 or 5 numbers in a row (that is, any row
parallel to an edge of the hexagon), the total is 38. this is the only
size magic hexagon that is possible.
39 = 3 × 13 = 3 + 5 + 7 + 11 + 13, the sum of a sequence of consecutive primes and the product of the first and last primes in that sequence. The first such sequence is 10 = 2 × 5 = 2 + 3 + 5; after 39 there are at least three more: 155, 371 and 454539357304421. (Related to Sloane's sequences A055233 and A055514).
Unaware of the "sum of sequence of primes" property, David Wells4 (in the first version of his book) called 39 the "smallest uninteresting positive integer", i.e. the first number with no particularly special attributes. (In the second edition of his book, that honor went to 51.) This is usually used as example of the type of contradictions one encounters when considering self-referential statements (if it's the first uninteresting number, then that makes it interesting, right?). I prefer to look at it as an example of the never-ending depth of study in a field: If you look hard enough and long enough, you can find something interesting about any number. In the case of 39, the "sum of primes" property mentioned above was found only in the last 50 years, which is relatively recent for such a small number. (Of course, 39 has been linked to 666, but so has everything else.)
My first uninteresting number is currently here.
The frequent Biblical references to 40 (40 days of rain, 40 years wandering in the wilderness, 40 days of Lent, etc.) probably result from 40 being a common idiom for "many".
By contrast, "40 acres and a mule" (the unfulfilled promise to African Americans by General Sherman) probably gets its "40" from the fact that 40 acres is 1/16 of a square mile.
Euler discovered that the polynomial x^2+x+41 gives a prime number for all x from 0 to 39. There was much speculation about whether some polynomial or similar formula could give primes for all values of its variables, and that was eventually proven impossible (Goldbach, 1752).
There are other, similar polynomials with smaller constant terms, for example x^2+x+17. They are related to the same set of numbers that cause near-integer values of eπ√n (see 163). This is more obvious if you consider that (for example) x^2+x+41=(x+1/2)2+163/4.
There are formulas that have multiple variables for which, if the value of the formula is positive, it is prime. There are even ones for which every prime can be found as the value of the formula for some set of positive values of the variables. Such a formula was found by J.P. Jones, Hideo Wada, Daihachiro Sato and Douglas Wiens, and published in 1976:
(k+2){1 - [wz+h+j-q]2 - [(gk+2g+k+1)(h+j) + h - z]2 -[2n+p+q+z-e]2 -[16(k+1)3(k+2)(n+1)2 + 1 - f2]2 - [e3(e+2)(a+1)2 + 1 - o2]2 - [(a2-1)y2 + 1 - x2]2 -[16r2y4(a2-1) + 1 - u2]2 - [((a + u2(u2-a))2 - 1) (n+4dy)2 + 1 - (x+cu)2]2 - [n+l+v-y]2 -[(a2-1)l2 + 1 - m2]2 - [ai+k+1-l-i]2 - [p + l(a-n-1) + b(2an+2a-n2-2n-2) - m]2 -[q + y(a-p-1) + s(2ap+2a-p2-2p-2) - x]2 -[z + pl(a-p) + t(2ap-p2-1) - pm]2}
The number of variables can be reduced to 10. However, that requires massively increasing the size of the exponents (possibly as high as 1.6×1045 — shown by J.P. Jones in 1982).
The Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ... (Sloane's A000108) are generated by the formula Cn=(2n)!/(n!×(n+1)!). The following illustrates the first few Catalan numbers (notice the similarities and differences to the Motzkin numbers):
There is a triangle similar to Pascal's triangle for which each row adds up to a Catalan number:
The Catalan numbers are also related to the Bell numbers.
"42 months" is mentioned twice in the book of Revelation; see 1260.
42 is a cult number, whose cult status definitely originated with Douglas Adams' use of 42 as "The answer to the ultimate question of life, the universe and everything" in his Hitchhiker's Guide to the Galaxy series. Most current pop cultures uses of 42 derive from this. (See also 2.74858523...×1080588).
Lewis Carroll also mentioned 42 a few times in his writings, but did not manage to make 42 into a cult phenomenon the way Adams has.
(Gratuitous connections with 27: 3×(2×7)=42 but 3×(2+7)=27; (42 + 4×2 + 4)/2 = 27.)
The sequence 2, 3, 7, 43, 1807, 3263443, 10650056950807, ... is generated by taking the product of the first n terms and adding 1 (for example, 2×3×7+1=43). This is called Sylvester's sequence, and is Sloane's sequence A000058. Because of the way the terms are calculated, no two terms in the sequence have a common factor. Because of that, no two terms have a common prime factor, and therefore there are an infinite number of primes. See also 47.
The factorials count the number of different ways n items can be arranged. The subfactorials, also called recontres numbers, count the number of ways they can be arranged such that none is in its "proper" place. For example, three things (A,B,C) can be arranged 6 ways, but only two (C,A,B) and (B,C,A) have none of the items in their "original" place. The subfactorials are: 0, 1, 2, 9, 44, 265, 1854, 14833, ... They can be generated by:
A1 = 0
An = n An-1 + -1n
or (curiously enough) by:
An = round(n!/e)
(the "round" function rounds to the nearest integer). Due to that formula, if you shuffle n objects randomly, there is about a 1/e chance that none of them will be in its original position. This in turn is related to the definition of e — the limit (as n approaches infinity) of (1+1/n)n.
| 45 | = 45 | 45 | = 45 |
| 452 | = 2025 | 20+25 | = 45 |
| 453 | = 91125 | 9+11+25 | = 45 |
| 454 | = 4100625 | 4+10+06+25 | = 45 |
Kaprekar numbers
452 = 2025, and 20+25 = 45. Numbers like this are called Kaprekar numbers. There are a few ways to define the sequence, depending on whether you allow dividing the square into two "pieces" of unequal size (for example, see 4879) and whether a trivial case like a power of 10 should count. I interpret the Kaprekar numbers the way they were originally described by Wells: If the original number N has D digits, then after squaring you should split it into a "right half" of D digits, equal to N2 mod 10D, and a "left half" of D or D-1 digits, equal to floor(N2/10D). Using those rules, the sequence of Kaprekar numbers runs: 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, ... (Sloane's A053816, more terms here)
The Kaprekar number concept can be extended to higher powers. 453= 091125, and 09+11+25=45. The numbers with this property are: 1, 8, (10), 45, 297, 2322, 2728, 4445, 4544, 4949, 5049, 5455, 5554, 7172, 27100, 44443, 55556, 60434, 77778, 143857, ... (more terms here). 10 is shown in parentheses because some folks consider it to be too trivial to count, however it does meet the requirements of the formula: 103=1000, break 1000 into groups of 2 digits starting from the right, 10+00=10.
45 is also a Kaprekar number for 4th powers: 454=04100625; 04+10+06+25=45. This sequence runs: 1, 7, 45, 55, 67, (100), 433, 4950, 5050, 38212, 65068, ... (more terms here])
45 is the only number (up to at least 400000) that is in all three of these Kaprekar sequences.
46 is the product of two distinct primes. This sequence, a subset of the semiprimes, begins: 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, ... (Sloane's A006881). A similar sequence requiring the two primes to be consecutive is discussed in the entry for 77.
The first of three consecutive primes that are spaced an equal distance apart: 47, 53 and 59 are spaced 6 apart and have no other primes in between. The first such set of 3 primes is (3, 5, 7); the next three are (151, 157, 163) and after that it becomes a bit more frequent. See also 251, 9843019, 121174811, 19252884016114523644357039386451 and 2.0014732742×1051089.
47 is a member of a sequence with a simple definition and a few interesting properties. The sequence begins with 3 and each successive term An+1 is An2-2. This is just like the definition of the Lucas-lehmer sequence but starting with 3 instead of 4. The sequence starts: 3, 7, 47, 2207, 4870847, 23725150497407, 562882766124611619513723647, ... (Sloane's A01566). Each term An is equal to the ratio F2n+2 / F2n+1 where Fn are the Fibonacci numbers (for example, F16/F8 = 987/21 = 47).
Also, every term is relatively prime to every other term, like the Sylvester sequence. This can be demonstrated by considering what happens when you show the terms of the sequence in base p, where p is any prime number. First consider base 10 — the first term is 3, and after that every term ends in 7. It is easy to see why — whenever you start with a number ending in 7 and calculate n2, you'll get a number ending in 9 — then after subtracting 2, it will end in 7 again. Here are the first 5 terms in two other bases:
| base | base 7 | base 11 |
| A1 | 37 | 311 |
| A2 | 107 | 711 |
| A3 | 657 | 4311 |
| A4 | 63027 | 172711 |
| A5 | 562545027 | 28275A311
|
Notice the final digits highlighted in bold. In base 7, the second term ends in 0, because the second term is divisible by 7. The next term ends in 5, which is 2 less than the base — because of the "minus 2" in the formula An2-2. The term after that ends in 2, because 52-2 is 23, which in base 7 is 327, ending in 2. It is true in any base that (b-2)2-2 = 2. And after that, all following terms will also end in the digit 2, because 22-2 = 2. Similarly, if there is any term that is divisible by the prime p, that term will end in 0 in base p, the following term will end in p-2 and all terms after that will end in 2. Therefore, only one term is divisible by p for any prime p, and it follows that every term is relatively prime to every other term.
The base 11 case shows that there are some prime numbers that divide into none of the terms of the sequence. In base 11, the final digit alternates between 3 and 7 forever.
Notable for having a lot of factors: 48 is 24×3 and is divisible by 2, 3, 4, 6, 8, 12, 16, and 24.
One of a sequence (Sloane's A033942) of numbers that have at least three prime factors (as with 48, the three primes do not need to be distinct, so for example 8=23 counts as the product of 3 primes). The sequence starts: 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, ... Some of the numbers in this sequence are divisible by three different primes. The composite numbers not in this sequence are called semiprimes.
48 and many of the smaller numbers in the A033942 sequence are also 3-smooth numbers.
49 = 7×7, the square of a traditionally "lucky" number. 49 is also "lucky" in another sense.
Stanislaw Ulam devised a sequence of numbers called lucky numbers. The sequence is generated by a sieve similar to the Sieve of Eratosthenes that (can be) used to generate the sequence of primes. The lucky numbers are: 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, ... (Sloane's A000959). These numbers share a few interesting properties with the primes, most of them similarities in statistical distribution.
50 = 12+72 = 52+52, the smallest number that is the sum of squares in two different ways (if the squares must be distinct, it would be 65). The numbers with this property are: 50, 65, 85, 125, 130, 145, 170, 185, 200, 205, 221, 250, 260, 265, 290, 305, 325, 338, 340, 365, 370, 377, ... (Sloane's A007692). See also 1729 and 635318657.
50 is also the sum of three consecutive squares: 32+42+52=50 (see 216).
Also a magic number in nuclear physics.
51 is a Motzkin number. These numbers count combinations of
things, in a way similar to the Bell and
Catalan numbers. In this case, they count certain orderings
of parentheses, or connected paths on a unit grid, or noninterseting
chords on a circle. The sequence starts: 1, 1, 2, 4,
9, 21, 51, 127, 323, 835, 2188, 5798, 15511,
41835, 113634, 310572, 853467, 2356779, 6536382, ... (Sloane's
A001006). The following illustrates the first few
Motzkin numbers (notice the similarities and differences
to the Catalan numbers):
Left side illustrates: Right side illustrates:
Number of paths of N steps Number of distinct ways
where each step goes east, parentheses can be substituted
northeast or southeast, and for X's in a string of N
never goes further south characters, with the
than the starting point. parentheses balanced.
A[1] = 1 _ x
A[2] = 2 __ /\ xx ()
_
A[3] = 4 ___ _/\ /\_ / \ xxx x() ()x (x)
_
____ __/\ _/\_ _/ \ xxxx xx() x()x x(x)
_ __
A[4] = 9 /\__ /\/\ / \_ / \ ()xx ()() (x)x (xx)
/\
/ \ (())
xxxxx xxx() xx()x xx(x)
x()xx x()() x(x)x x(xx)
A[5] = 21 x(()) ()xxx ()x() ()()x
()(x) (x)xx (x)() (xx)x
(xxx) (x()) (())x (()x)
((x))
A[6] = 51 xxxxxx xxxx() xxx()x xxx(x) xx()xx xx()() xx(x)x xx(xx)
xx(()) x()xxx x()x() x()()x x()(x) x(x)xx x(x)() x(xx)x
x(xxx) x(x()) x(())x x(()x) x((x)) ()xxxx ()xx() ()x()x
()x(x) ()()xx ()()() ()(x)x ()(xx) ()(()) (x)xxx (x)x()
(x)()x (x)(x) (xx)xx (xx)() (xxx)x (xxxx) (xx()) (x())x
(x()x) (x(x)) (())xx (())() (()x)x (()xx) (()()) ((x))x
((x)x) ((xx)) ((()))
One of the Bell numbers: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ... (Sloane's A000110). These numbers count the number of ways of putting n distinguishable balls into one or more indistinguishable urns. For example:
1: ( 1 )
2: ( 1 2 ) ( 2 )( 1 )
5: ( 1 2 3 ) ( 32 )( 1 ) ( 31 )( 2 ) ( 21 )( 3 )
( 3 )( 2 )( 1 )
15: ( 1 2 3 4 ) ( 4 3 2 )( 1 ) ( 4 3 1 )( 2 ) ( 4 2 1 )( 3 )
( 3 2 1 )( 4 ) ( 4 3 )( 2 1 ) ( 4 3 )( 2 )( 1 )
( 4 2 )( 3 1 ) ( 4 2 )( 3 )( 1 ) ( 4 1 )( 3 2 )
( 4 1 )( 3 )( 2 ) ( 3 2 )( 4 )( 1 ) ( 3 1 )( 4 )( 2 )
( 2 1 )( 4 )( 3 ) ( 4 )( 3 )( 2 )( 1 )
If the urns are distinguishable, then we get the so-called ordered Bell numbers. If the balls and urns are both indistinguishable, we get the partition numbers. The Bell numbers are also related to the Catalan numbers.
The set of combinations is equivalent to what is happening in the 52 different patterns of genjiko. In this traditional Japanese game, 5 incense sticks are selected at random from a supply that includes 5 varieties. The sticks are laid vertically side-by-side and lit; the player then draws a diagram of five vertical lines, and connects the tops of any lines that are found to be the same.
The Bell numbers can be generated by the following table, which works kind of like Pascal's Triangle. Write the known terms of the sequence across the top, and add together, for example 7+20=27, and at the bottom you will get the next term in the series:
| 1 | 1 | 2 | 5 | 15 | ... | ||||
| 2 | 3 | 7 | 20 | ||||||
| 5 | 10 | 27 | |||||||
| 15 | 37 | ||||||||
| 52 |
It is possible to arrange all the natural numbers into a sequence of fractions like this:
(1+2)/3
(4+5+6+7+8+9+10+11)/12
(13+14+15+ ... +36+37+38)/39
(40+41+42+ ... +118+119)/120
...
so that each fraction is a whole number. When this is done, the denominators are all numbers of the form (3n-3)/2, and the fractions are of the form 2×3n-1: 1, 5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, ... (Sloane's A048473; my MCS125664)
54 is the number of colored squares on a Rubik's CubeTM or on similar toys and puzzles that take the form of a cube with 3×3 pattern on each face. This is also twice the number of "little cubes" (33×2) but that is just a coincidence.
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© 1996-2008 Robert P. Munafo.
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