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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. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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.

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

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.

EXAMPLE

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].

PROG

(Scheme)

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

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.

Sequence in context: A275837 A119510 A290095 * A242528 A137933 A143116

Adjacent sequences:  A275722 A275723 A275724 * A275726 A275727 A275728

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 09 2016

STATUS

approved

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Last modified July 12 10:20 EDT 2020. Contains 335657 sequences. (Running on oeis4.)