Notable Properties of Specific Numbers

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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 10^D, and a "left half" of D or D-1 digits, equal to floor(N²/10^D). 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 A53816, 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 4^th 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.

Distinct Semiprimes

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 A6881). 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 A_n+1 is A_n²-2. 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 A1566). Each term A_n is equal to the ratio F_2ⁿ⁺² / F_2ⁿ⁺¹ where F_n are the Fibonacci numbers (for example, F₁₆/F₈ = 987/21 = 47).

Also, every term is relatively prime to every other term, something I call mutually co-prime. (The Sylvester sequence is another example). 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 A_n²-2. 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. Thus, no terms are divisible by 11.

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 A33942) 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.

Lucky Numbers

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 A0959). 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 A7692). 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 A1006). 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)) ((()))

52 is one of the Bell numbers: 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, ... (Sloane's A0110). 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 A48473; my MCS125664)

54 is the number of colored squares on a Rubik's Cube^TM 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.

The number that most Americans know best from its use as a speed limit also happens to be a Fibonacci number and the sum of the numbers 1 through 10 (which makes it the 10th triangular number). There are no larger numbers that are both a Fibonacci and triangular.

55 is also 5²+4²+3²+2²+1². Such numbers are called pyramidal numbers, because you can stack the squares on top of each other (with the largest on the bottom) to make a pyramid. Sometimes they are called square pyramidal numbers to distinguish them from the tetrahedral numbers. You can add the next square to each pyramidal number to get the next pyramidal number. The pyramidal numbers are: 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, ... (Sloane's A0330; my MCS1936).

Another way to express the sum of squares above is: 5×5+4×4+3×3+2×2+1×1. If you change the + and × in that expression to × and exponents respectively, you get 86400000, the number of milliseconds in a day.

56 is one of the partition numbers, which count the number of ways of placing N indistinguishable balls into N indistinguishable urns (but with no empty urns). For example:

1: (*)
2: (**) (*)(*)
3: (***) (**)(*) (*)(*)(*)
5: (****) (***)(*) (**)(**) (**)(*)(*) (*)(*)(*)(*)
7: (*****) (****)(*) (***)(**) (***)(*)(*) (**)(**)(*) (**)(*)(*)(*) (*)(*)(*)(*)(*)
11: (******) (*****)(*) (****)(**) (****)(*)(*) (***)(***) (***)(**)(*) (***)(*)(*)(*) (**)(**)(**) (**)(**)(*)(*) (**)(*)(*)(*)(*) (*)(*)(*)(*)(*)(*)

The sequence runs: 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, ... (Sloane's A0041, more terms here).

If the urns are distinguishable, then we get the powers of two. If the balls are distinguishable, we get the Bell numbers.

Latin Squares

There are 161280 ways to arrange the digits 1 through 5 in a 5×5 square so that each digit appears once in each row and once in each column. Such an arrangement is called a latin square. If the first row has its digits in increasing order, it is called a reduced latin square. There are 56 5×5 reduced latin squares. Here is an example:

1 2 3 4 5
2 3 5 1 4
3 5 4 2 1
4 1 2 5 3
5 4 1 3 2

9×9 latin squares are familiar to anyone who has played sudoku.

56.9612484322608203...

This is the value of x for which x^x=10¹⁰⁰. It is therefore the biggest number you can raise to its own power on most scientific calculators. See also 2.506184... and 69.

57 can be expressed as a sum x^y + y^x where x and y are integers greater than 1 — in this case, 2⁵+5²=57. Such numbers are called Leyland numbers; the sequence begins: 8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945, 1124, 1649, 2169, 2530, 4240, 5392, 6250, 7073, 8361, ... (Sloane's A76980. Here is a table showing the 22 Leyland numbers less than 10000:

	x=2	x=3	x=4	x=5
y=2	8
y=3	17	54
y=4	32	145	512
y=5	57	368	1649	6250
y=6	100	945	5392
y=7	177	2530
y=8	320	7073
y=9	593
y=10	1124
y=11	2169
y=12	4240
y=13	8361