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 A005225 Number of permutations of length n with equal cycles. (Formerly M0903) 15
 1, 2, 3, 10, 25, 176, 721, 6406, 42561, 436402, 3628801, 48073796, 479001601, 7116730336, 88966701825, 1474541093026, 20922789888001, 400160588853026, 6402373705728001, 133991603578884052, 2457732174030848001, 55735573291977790576, 1124000727777607680001 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS a(n) = (n-1)! + 1 iff n is a prime. REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). D. P. Walsh, A differentiation-based characterization of primes, Abstracts Amer. Math. Soc., 25 (No. 2, 2002), p. 339, #975-11-237. LINKS Alois P. Heinz, Table of n, a(n) for n = 1..200 R. K. Guy, Letter to N. J. A. Sloane, Jul 1988 D. P. Walsh, Primality test based on the generating function D. P. Walsh, A differentiation-based characterization of primes H. S. Wilf, Three problems in combinatorial asymptotics, J. Combin. Theory, A 35 (1983), 199-207. FORMULA a(n) = n!*sum(((n/k)!*k^(n/k))^(-1)) where sum is over all divisors k of n. Exponential generating function [for a(1) through a(n)]= sum(exp(t^k/k)-1, k=1..n). EXAMPLE For example, a(4)=10 since, of the 24 permutations of length 4, there are 6 permutations with consist of a single 4-cycle, 3 permutations that consist of two 2-cycles and 1 permutation with four 1-cycles. Also, a(7)=721 since there are 720 permutations with a single cycle of length 7 and 1 permutation with seven 1-cycles. MAPLE with(numtheory): a:= n-> n!*add((d!*(n/d)^d)^(-1), d=divisors(n)): seq(a(n), n=0..30);  # Alois P. Heinz, Nov 07 2012 MATHEMATICA Table[n! Sum[((n/d)!*d^(n/d))^(-1), {d, Divisors[n]}], {n, 21}] (* Jean-François Alcover, Apr 04 2011 *) PROG (Maxima) a(n):= n!*lsum((d!*(n/d)^d)^(-1), d, listify(divisors(n))); makelist(a(n), n, 1, 40); /* Emanuele Munarini, Feb 03 2014 */ CROSSREFS Cf. A038041, A055225, A236696, A317329. Column k=1 of A218868. Sequence in context: A341265 A005158 A182926 * A211208 A303836 A238937 Adjacent sequences:  A005222 A005223 A005224 * A005226 A005227 A005228 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Additional comments from Dennis P. Walsh, Dec 08 2000 More terms from Vladeta Jovovic, Dec 01 2001 STATUS approved

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Last modified June 15 11:44 EDT 2021. Contains 345048 sequences. (Running on oeis4.)