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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.
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%I #34 Jun 24 2017 09:14:00

%S 2,4,18,8,12,8,150,100,54,16,24,16,90,40,54,16,36,16,60,40,36,16,24,

%T 16,1470,980,882,392,588,392,750,500,162,32,48,32,270,80,162,32,108,

%U 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

%N 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.

%C 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.

%H Antti Karttunen, <a href="/A275725/b275725.txt">Table of n, a(n) for n = 0..40319</a>

%H Indranil Ghosh, <a href="/A275725/a275725.txt">Python program for computing this sequence</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

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

%F Other identities:

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

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

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

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

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

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

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

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

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

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

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

%e Consider the first eight permutations (indices 0-7) listed in A060117:

%e 1 [Only the first 1-cycle explicitly listed thus a(0) = 2^1 = 2]

%e 2,1 [One transposition (2-cycle) in beginning, thus a(1) = 2^2 = 4]

%e 1,3,2 [One fixed element in beginning, then transposition, thus a(2) = 2^1 * 3^2 = 18]

%e 3,1,2 [One 3-cycle, thus a(3) = 2^3 = 8]

%e 3,2,1 [One transposition jumping over a fixed element, a(4) = 2^2 * 3^1 = 12]

%e 2,3,1 [One 3-cycle, thus a(5) = 2^3 = 8]

%e 1,2,4,3 [Two 1-cycles, then a 2-cycle, thus a(6) = 2^1 * 3^1 * 5^2 = 150].

%e 2,1,4,3 [Two 2-cycles, not crossed, thus a(7) = 2^2 * 5^2 = 100]

%e and also the seventeenth one at n=16 [A007623(16)=220] where we have:

%e 3,4,1,2 [Two 2-cycles crossed, thus a(16) = 2^2 * 3^2 = 36].

%o (Scheme)

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

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

%Y Cf. A275807 (terms divided by 2).

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

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

%K nonn

%O 0,1

%A _Antti Karttunen_, Aug 09 2016