A181955 -id:A181955 - cp's OEIS frontend

A181951 Number of cyclic subgroups of prime order in the Alternating Group A_n.

0, 0, 1, 7, 31, 121, 526, 2227, 9283, 54931, 694156, 6104011, 76333687, 872550043, 7491293356, 49469173951, 1571562887071, 24729107440927, 584036983443568, 8662243014551731, 87570785839885951, 1147293350653737211, 66175018194591458692, 1378758190497550145383

Offset: 1

Olivier Gérard, Apr 03 2012

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A001465, A181955, A186202 (symmetric group).

a[n_] := Sum[If[PrimeQ[p], Sum[If[p > 2 || Mod[k, 2] == 0, n!/(k!*(n - k*p)!*p^k)/(p - 1), 0], {k, 1, n/p}], 0], {p, 2, n}];
Array[a, 24] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, if(p>2||k%2==0, n!/(k!*(n-k*p)!*p^k)))/(p-1)))}

a(n) = A186202(n) - A001465(n). - Andrew Howroyd, Jul 04 2018

Terms a(9) and beyond from Andrew Howroyd, Jul 04 2018

A181954 Weighted sum of all cyclic subgroups of prime order of the symmetric group S_n.

Original entry on oeis.org

0, 2, 9, 30, 110, 450, 2457, 11774, 47910, 264630, 5565065, 56021922, 866143434, 9894742130, 78233264865, 470058202350, 24530748587102, 409761902222094, 10595012400106545, 160826238368038490, 1585844131838898330, 16787211702213659322, 1362379222505265018329

Offset: 1

Olivier Gérard, Apr 03 2012

nonn

Sum of the orders of all subgroups of prime order in S_n.

			The symmetric group S_3 has one subgroup of order 3 and three subgroups of order 2, and no other subgroups of prime order. So a(3) = 3 + 2 + 2 + 2 = 9.

Stephen A. Silver, Table of n, a(n) for n = 1..451

Cf. A181955 (the alternating case).

a[n_] := Sum[If[PrimeQ[p], Sum[n!/(k!*(n-k*p)!*p^k), {k, 1, n/p}]*p/(p-1), 0], {p, 2, n}];
Array[a, 24] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

a(n)={sum(p=2, n, if(isprime(p), sum(k=1, n\p, n!/(k!*(n-k*p)!*p^k))*p/(p-1)))} \\ Andrew Howroyd, Jul 03 2018

a(n) = A186202(n) + A214003(n).

More terms from Stephen A. Silver, Feb 16 2013

cp's OEIS Frontend

A181951 Number of cyclic subgroups of prime order in the Alternating Group A_n.

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