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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
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.
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. A275807 (terms divided by 2).
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 09 2016
STATUS
approved