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