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 A060014 Sum of orders of all permutations of n letters. 13
 1, 1, 3, 13, 67, 471, 3271, 31333, 299223, 3291487, 39020911, 543960561, 7466726983, 118551513523, 1917378505407, 32405299019941, 608246253790591, 12219834139189263, 253767339725277823, 5591088918313739017, 126036990829657056711, 2956563745611392385211 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: This sequence eventually becomes cyclic mod n for all n. - Isaac Saffold, Dec 01 2019 REFERENCES D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.2, p. 460. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..170 FindStat - Combinatorial Statistic Finder, The order of a permutation Joshua Harrington, Lenny Jones, and Alicia Lamarche, Characterizing Finite Groups Using the Sum of the Orders of the Elements, International Journal of Combinatorics, Volume 2014, Article ID 835125, 8 pages. FORMULA E.g.f.: Sum_{n>0} (n*Sum_{i|n} (moebius(n/i)*Product_{j|i} exp(x^j/j))). - Vladeta Jovovic, Dec 29 2004; The sum over n should run to at least A000793(k) for producing the k-th entry. - Wouter Meeussen, Jun 16 2012 a(n) = Sum_{k>=1} k* A057731(n,k). - R. J. Mathar, Aug 31 2017 EXAMPLE For n = 4 there is 1 permutation of order 1, 9 permutations of order 2, 8 of order 3 and 6 of order 4, for a total of 67. MAPLE b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!       *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))     end: a:= n-> b(n, 1): seq(a(n), n=0..30);  # Alois P. Heinz, Jul 11 2017 MATHEMATICA CoefficientList[Series[Sum[n Fold[#1+MoebiusMu[n/#2] Apply[Times, Exp[x^#/#]&/@Divisors[#2] ]&, 0, Divisors[n]], {n, Max[Apply[LCM, Partitions[19], 1]]}], {x, 0, 19}], x] Range[0, 19]! (* Wouter Meeussen, Jun 16 2012 *) a[ n_] := If[ n < 1, Boole[n == 0], 1 + Total @ Apply[LCM, Map[Length, First /@ PermutationCycles /@ Drop[Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *) PROG (PARI) \\ Naive method -- sum over cycles directly cycleDecomposition(v:vec)={     my(cyc=List(), flag=#v+1, n);     while((n=vecmin(v))<#v,         my(cur=List(), i, tmp);         while(v[i++]!=n, );         while(v[i] != flag,             listput(cur, tmp=v[i]);             v[i]=flag;             i=tmp         );         if(#cur>1, listput(cyc, Vec(cur)))    \\ Omit length-1 cycles     );     Vec(cyc) }; permutationOrder(v:vec)={     lcm(apply(length, cycleDecomposition(v))) }; a(n)=sum(i=0, n!-1, permutationOrder(numtoperm(n, i))) \\ Charles R Greathouse IV, Nov 06 2014 (PARI) A060014(n) = {   my(factn = n!, part, nb, i, j, res = 0);   forpart(part = n,     nb = 1; j = 1;     for(i = 1, #part,       if (i == #part || part[i + 1] != part[i],         nb *= (i + 1 - j)! * part[i]^(i + 1 - j);         j = i + 1));     res += (factn / nb) * lcm(Vec(part)));   res; } \\ Jerome Raulin, Jul 11 2017 (much faster, O(A000041(n)) vs O(n!)) CROSSREFS Cf. A000793, A028418, A060015, A057731, A074859, A290932. Sequence in context: A080832 A194019 A020017 * A182666 A042659 A054132 Adjacent sequences:  A060011 A060012 A060013 * A060015 A060016 A060017 KEYWORD nonn,nice,easy AUTHOR N. J. A. Sloane, Mar 17 2001 EXTENSIONS More terms from Vladeta Jovovic, Mar 18 2001 More terms from Alois P. Heinz, Feb 14 2013 STATUS approved

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Last modified December 2 07:16 EST 2021. Contains 349437 sequences. (Running on oeis4.)