# Speciality of Numbers

**Topics:**Integer sequences, Prime number, Figurate numbers

**Pages:**161 (57860 words)

**Published:**October 24, 2012

0 is the additive identity.

1 is the multiplicative identity.

2 is the only even prime.

3 is the number of spatial dimensions we live in.

4 is the smallest number of colors sufficient to color all planar maps. 5 is the number of Platonic solids.

6 is the smallest perfect number.

7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass. 8 is the largest cube in the Fibonacci sequence.

9 is the maximum number of cubes that are needed to sum to any positive integer. 10 is the base of our number system.

11 is the largest known multiplicative persistence.

12 is the smallest abundant number.

13 is the number of Archimedian solids.

14 is the smallest even number n with no solutions to φ(m) = n. 15 is the smallest composite number n with the property that there is only one group of order n. 16 is the only number of the form xy = yx with x and y being different integers. 17 is the number of wallpaper groups.

18 is the only positive number that is twice the sum of its digits. 19 is the maximum number of 4th powers needed to sum to any number. 20 is the number of rooted trees with 6 vertices.

21 is the smallest number of distinct squares needed to tile a square. 22 is the number of partitions of 8.

23 is the smallest number of integer-sided boxes that tile a box so that no two boxes share a common length. 24 is the largest number divisible by all numbers less than its square root. 25 is the smallest square that can be written as a sum of 2 squares. 26 is the only positive number to be directly between a square and a cube. 27 is the largest number that is the sum of the digits of its cube. 28 is the 2nd perfect number.

29 is the 7th Lucas number.

30 is the largest number with the property that all smaller numbers relatively prime to it are prime. 31 is a Mersenne prime.

32 is the smallest non-trivial 5th power.

33 is the largest number that is not a sum of distinct triangular numbers. 34 is the smallest number with the property that it and its neighbors have the same number of divisors. 35 is the number of hexominoes.

36 is the smallest non-trivial number which is both square and triangular. 37 is the maximum number of 5th powers needed to sum to any number. 38 is the last Roman numeral when written lexicographically. 39 is the smallest number which has 3 different partitions into 3 parts with the same product. 40 is the only number whose letters are in alphabetical order. 41 is a value of n so that x2 + x + n takes on prime values for x = 0, 1, 2, ... n-2. 42 is the 5th Catalan number.

43 is the number of sided 7-iamonds.

44 is the number of derangements of 5 items.

45 is a Kaprekar number.

46 is the number of different arrangements (up to rotation and reflection) of 9 non-attacking queens on a 9×9 chessboard. 47 is the largest number of cubes that cannot tile a cube. 48 is the smallest number with 10 divisors.

49 is the smallest number with the property that it and its neighbors are squareful. 50 is the smallest number that can be written as the sum of of 2 squares in 2 ways. 51 is the 6th Motzkin number.

52 is the 5th Bell number.

53 is the only two digit number that is reversed in hexadecimal. 54 is the smallest number that can be written as the sum of 3 squares in 3 ways. 55 is the largest triangular number in the Fibonacci sequence. 56 is the number of reduced 5×5 Latin squares.

57 = 111 in base 7.

58 is the number of commutative semigroups of order 4.

59 is the number of stellations of an icosahedron.

60 is the smallest number divisible by 1 through 6.

61 is the 3rd secant number.

62 is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways. 63 is the number of partially ordered sets of 5 elements. 64 is the smallest number with 7 divisors.

65 is the smallest number that becomes square if its reverse is either added to or subtracted from it. 66 is the number of 8-iamonds.

67 is...

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