A275725 a(n) = A275723(A002110(1+A084558(n)), n); prime factorization encodings of cycle-polynomials computed for finite permutations listed in the order that is used in tables A060117 / A060118.

2, 4, 18, 8, 12, 8, 150, 100, 54, 16, 24, 16, 90, 40, 54, 16, 36, 16, 60, 40, 36, 16, 24, 16, 1470, 980, 882, 392, 588, 392, 750, 500, 162, 32, 48, 32, 270, 80, 162, 32, 108, 32, 120, 80, 72, 32, 48, 32, 1050, 700, 378, 112, 168, 112, 750, 500, 162, 32, 48, 32, 450, 200, 162, 32, 72, 32, 300, 200, 108, 32, 48, 32, 630, 280, 378, 112, 252, 112, 450, 200

Offset: 0

Author

Antti Karttunen, Aug 09 2016

Keywords

• nonn

Comments

In this context "cycle-polynomials" are single-variable polynomials where the coefficients (encoded with the exponents of prime factorization of n) are equal to the lengths of cycles in the permutation listed with index n in tables A060117 or A060118. See the examples.

Examples

			Consider the first eight permutations (indices 0-7) listed in A060117:
  1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]
  2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]
  1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]
  3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]
  3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]
  2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]
  1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].
  2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]
and also the seventeenth one at n=16 [A007623(16)=220] where we have:
  3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..40319
• Indranil Ghosh, Python program for computing this sequence
• Index entries for sequences related to factorial base representation

Crossrefs

Cf. A000040, A001222, A001221, A002110, A007814, A046660, A048675, A051903, A056169, A056170, A084558, A243054, A257510, A275723, A275803, A275832.

Cf. A275807 (terms divided by 2).

Cf. also A060129, A060128, A060130, A060131, A072411, A275726, A275812, A275851.

Cf. also A275733, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

Programs

Scheme

(define (A275725 n) (A275723bi (A002110 (+ 1 (A084558 n))) n)) ;; Code for A275723bi given in A275723.

Formula

a(n) = A275723(A002110(1+A084558(n)), n).

Other identities:

A001221(a(n)) = 1+A257510(n) (for all n >= 1).

A001222(a(n)) = 1+A084558(n).

A007814(a(n)) = A275832(n).

A048675(a(n)) = A275726(n).

A051903(a(n)) = A275803(n).

A056169(a(n)) = A275851(n).

A046660(a(n)) = A060130(n).

A072411(a(n)) = A060131(n).

A056170(a(n)) = A060128(n).

A275812(a(n)) = A060129(n).

a(n!) = 2 * A243054(n) = A000040(n)*A002110(n) for all n >= 1.