OFFSET
1,3
LINKS
Stephen A. Silver, Table of n, a(n) for n = 1..451
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
EXAMPLE
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.
MAPLE
b:= proc(n, p) option remember;
`if`(n<p, 0, b(n-1, p)+(1+b(n-p, p))*(n-1)!/(n-p)!)
end:
a:= 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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Stephen A. Silver, Feb 15 2013
STATUS
approved