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URL: https://oeis.org/A214003

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A214003
Number of degree-n permutations of prime order.
4
0, 1, 5, 17, 69, 299, 1805, 9099, 37331, 205559, 4853529, 49841615, 789513659, 9021065871, 70737031469, 420565124399, 22959075244095, 385032305178719, 10010973102879761, 152163983393187399, 1498273284120348539, 15639918041915598815, 1296204202723400597109
OFFSET
1,3
LINKS
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
# Alternative:
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 *)
KEYWORD
nonn
AUTHOR
Stephen A. Silver, Feb 15 2013
STATUS
approved