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A214003 Number of degree-n permutations of prime order.

%I A214003 #23 Jan 19 2023 16:51:46
%S A214003 0,1,5,17,69,299,1805,9099,37331,205559,4853529,49841615,789513659,
%T A214003 9021065871,70737031469,420565124399,22959075244095,385032305178719,
%U A214003 10010973102879761,152163983393187399,1498273284120348539,15639918041915598815,1296204202723400597109
%N A214003 Number of degree-n permutations of prime order.
%H A214003 Stephen A. Silver, <a href="/A214003/b214003.txt">Table of n, a(n) for n = 1..451</a>
%F A214003 a(n) = Sum_{p prime} A057731(n,p).
%F A214003 E.g.f.: exp(x)*Sum_{p in Primes} exp(x^p/p)-1. - _Geoffrey Critzer_, Nov 08 2015
%e A214003 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.
%p A214003 b:= proc(n,p) option remember;
%p A214003       `if`(n<p, 0, b(n-1,p)+(1+b(n-p,p))*(n-1)!/(n-p)!)
%p A214003     end:
%p A214003 a:= n-> add(b(n, ithprime(i)), i=1..numtheory[pi](n)):
%p A214003 seq(a(n), n=1..30);  # _Alois P. Heinz_, Feb 16 2013
%p A214003 # second Maple program:
%p A214003 b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
%p A214003       add(b(n-j, ilcm(j, g))*(n-1)!/(n-j)!, j=1..n))
%p A214003     end:
%p A214003 a:= n-> b(n, 1):
%p A214003 seq(a(n), n=1..23);  # _Alois P. Heinz_, Jan 19 2023
%t A214003 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 *)
%Y A214003 Cf. A001189, A001471, A057731, A059593, A153760, A153761, A186202.
%K A214003 nonn
%O A214003 1,3
%A A214003 _Stephen A. Silver_, Feb 15 2013