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

%I A329944 #20 Jul 25 2023 11:03:27
%S A329944 1,1,1,1,7,31,211,1051,10081,107857,1227241,8969401,108817831,
%T A329944 1173362191,19426473067,320062090531,5692838161921,70426164947041,
%U A329944 1346222143950481,21952313047471537,493701484264143751,10971915198235355071,266542798822750395331
%N A329944 Number of permutations of [n] whose cycle lengths avoid primes.
%H A329944 Alois P. Heinz, <a href="/A329944/b329944.txt">Table of n, a(n) for n = 0..450</a>
%F A329944 a(n) mod 2 = 1.
%F A329944 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).
%e A329944 a(4) = 7: (1)(2)(3)(4), (1234), (1243), (1324), (1342), (1423), (1432).
%p A329944 a:= proc(n) option remember; `if`(n=0, 1, add(`if`(isprime(j), 0,
%p A329944       a(n-j)*binomial(n-1, j-1)*(j-1)!), j=1..n))
%p A329944     end:
%p A329944 seq(a(n), n=0..25);
%t A329944 a[n_] := a[n] = If[n == 0, 1, Sum[If[PrimeQ[j], 0,
%t A329944      a[n-j] Binomial[n-1, j-1] (j-1)!], {j, 1, n}]];
%t A329944 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Oct 31 2021, after _Alois P. Heinz_ *)
%Y A329944 Cf. A000040, A218002, A329945, A364450.
%K A329944 nonn
%O A329944 0,5
%A A329944 _Alois P. Heinz_, Nov 24 2019