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Number Theory

The functions in this section provide tools for number-theoretic computations:
divisor functions, partitions, polygonal numbers, perfect/happy numbers, and special combinatorial counts (Eulerian, Stirling).

Function Definitions

Totient: (n: integer) -> integer

Returns Euler’s totient function

φ(n)

: the number of integers

1 \leq k \leq n

that are coprime to

n

.

This function counts how many integers up to $n$ share no common factors with $n$ other than 1. For example, for $n=9$ , the integers coprime to 9 are 1, 2, 4, 5, 7, and 8, so $φ(9) = 6$ .

See also: Euler's totient function - Wikipedia

["Totient", 9]
// ➔ 6

Sigma0: (n: integer) -> integer

Returns the number of positive divisors of

n

.

This function counts how many positive integers divide $n$ without a remainder. For example, for $n=6$ , the divisors are 1, 2, 3, and 6, so the count is 4.

See also: Divisor function - Wikipedia

["Sigma0", 6]
// ➔ 4

Sigma1: (n: integer) -> integer

Returns the sum of positive divisors of

n

.

This function sums all positive divisors of $n$ . For example, for $n=6$ , the divisors are 1, 2, 3, and 6, and their sum is 12.

See also: Divisor function - Wikipedia

["Sigma1", 6]
// ➔ 12

SigmaMinus1: (n: integer) -> number

Returns the sum of reciprocals of the positive divisors of

n

.

This function sums the reciprocals of all positive divisors of $n$ . For example, for $n=6$ , the divisors are 1, 2, 3, and 6, and the sum of reciprocals is 1 + 1/2 + 1/3 + 1/6 = 2.333...

See also: Divisor function - Wikipedia

["SigmaMinus1", 6]
// ➔ 2.333...

Eulerian: (n: integer, m: integer) -> integer

Returns the Eulerian number A(n, m), counting permutations of with exactly m ascents.

Eulerian numbers count the permutations of the numbers 1 through $n$ that have exactly $m$ ascents (positions where the next number is greater than the previous one). For example, for $n=4$ and $m=2$ , there are 11 such permutations.

See also: Eulerian number - Wikipedia

["Eulerian", 4, 2]
// ➔ 11

Stirling: (n: integer, m: integer) -> integer

Returns the Stirling number of the second kind

S(n, m)

, counting the number of ways to partition

n

elements into

m

non-empty subsets.

Stirling numbers of the second kind count how many ways to split a set of $n$ objects into $m$ groups, none empty. For example, with $n=5$ and $m=2$ , there are 15 ways.

See also: Stirling number of the second kind - Wikipedia

["Stirling", 5, 2]
// ➔ 15

NPartition: (n: integer) -> integer

Returns the number of integer partitions of

n

.

This function counts how many ways $n$ can be expressed as a sum of positive integers, disregarding order. For example, 5 can be partitioned into 7 distinct sums: 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, and 1+1+1+1+1.

See also: Partition function (number theory) - Wikipedia

["NPartition", 5]
// ➔ 7

IsTriangular: (n: integer) -> boolean

Returns "True" if

n

is a triangular number.

Triangular numbers count objects arranged in an equilateral triangle. The $k$ th triangular number is $k(k+1)/2$ . For example, 15 is triangular because $5 \times 6 / 2 = 15$ .

See also: Triangular number - Wikipedia

["IsTriangular", 15]
// ➔ "True"

IsSquare: (n: integer) -> boolean

Returns "True" if

n

is a perfect square.

A perfect square is an integer that is the square of another integer. For example, 16 is a perfect square since $4^2 = 16$ .

See also: Square number - Wikipedia

["IsSquare", 16]
// ➔ "True"

IsPentagonal: (n: integer) -> boolean

Returns "True" if

n

is a pentagonal number.

Pentagonal numbers represent dots forming a pentagon. The $k$ th pentagonal number is given by $k(3k-1)/2$ . For example, 22 is pentagonal because it matches the formula for some integer $k$ .

See also: Pentagonal number - Wikipedia

["IsPentagonal", 22]
// ➔ "True"

IsOctahedral: (n: integer) -> boolean

Returns "True" if

n

is an octahedral number.

Octahedral numbers count objects arranged in an octahedron shape. The $k$ th octahedral number is given by $k(2k^2 + 1)/3$ . For example, 19 is not octahedral.

See also: Octahedral number - Wikipedia

["IsOctahedral", 19]
// ➔ "False"

IsCenteredSquare: (n: integer) -> boolean

Returns "True" if

n

is a centered square number.

Centered square numbers count dots arranged in a square with a dot in the center. The $k$ th centered square number is $(2k+1)^2$ . For example, 25 is centered square as it equals $5^2$ .

See also: Centered square number - OEIS A001844

["IsCenteredSquare", 25]
// ➔ "True"

IsPerfect: (n: integer) -> boolean

Returns "True" if

n

is a perfect number.

A perfect number is one that equals the sum of its positive divisors excluding itself. For example, 28 is perfect since 1 + 2 + 4 + 7 + 14 = 28.

See also: Perfect number - Wikipedia

["IsPerfect", 28]
// ➔ "True"

IsHappy: (n: integer) -> boolean

Returns "True" if

n

is a happy number.

A happy number is defined by iterating the sum of the squares of its digits; if this process eventually reaches 1, the number is happy. For example, 19 is happy because: 1²+9²=82; 8²+2²=68; 6²+8²=100; 1²+0²+0²=1.

See also: Happy number - Wikipedia

["IsHappy", 19]
// ➔ "True"

IsAbundant: (n: integer) -> boolean

Returns "True" if

n

is an abundant number.

An abundant number is one where the sum of its proper divisors exceeds the number itself. For example, 12 is abundant since 1 + 2 + 3 + 4 + 6 = 16 > 12.

See also: Abundant number - Wikipedia

["IsAbundant", 12]

Function Definitions