A214003 Number of degree-n permutations of prime order.

0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109

Offset: 1

Author

Stephen A. Silver, Feb 15 2013

Keywords

• nonn

Examples

			The symmetric group S_5 has 25 elements of order 2, 20 elements of order 3, and 24 elements of order 5. All other elements are of nonprime order (1, 4, or 6), so a(5) = 25 + 20 + 24 = 69.
		

Links

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

Crossrefs

Cf. A001189, A001471, A057731, A059593, A153760, A153761, A186202.

Programs

Maple

b:= proc(n,p) option remember;
      `if`(n add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 16 2013
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Jan 19 2023

Mathematica

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n],PrimeQ[Apply[LCM, #]] &]]], {n, 1,23}] (* Geoffrey Critzer, Nov 08 2015 *)

Formula

a(n) = Sum_{p prime} A057731(n,p).

E.g.f.: exp(x)*Sum_{p in Primes} exp(x^p/p)-1. - Geoffrey Critzer, Nov 08 2015