A329944 Number of permutations of [n] whose cycle lengths avoid primes.

1, 1, 1, 1, 7, 31, 211, 1051, 10081, 107857, 1227241, 8969401, 108817831, 1173362191, 19426473067, 320062090531, 5692838161921, 70426164947041, 1346222143950481, 21952313047471537, 493701484264143751, 10971915198235355071, 266542798822750395331

Offset: 0

Author

Alois P. Heinz, Nov 24 2019

Keywords

• nonn

Examples

			a(4) = 7: (1)(2)(3)(4), (1234), (1243), (1324), (1342), (1423), (1432).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000040, A218002, A329945, A364450.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j), 0,
      a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(a(n), n=0..25);

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[If[PrimeQ[j], 0,
     a[n-j] Binomial[n-1, j-1] (j-1)!], {j, 1, n}]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 31 2021, after Alois P. Heinz *)

Formula

a(n) mod 2 = 1.

a(n) mod 10 = period 5: repeat [1,1,1,1,7], g.f.: (7*x^4+x^3+x^2+x+1)/(1-x^5).