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A341462 Number of partitions of n into 4 distinct nonprime parts.

%I A341462 #12 Nov 07 2021 11:33:57
%S A341462 1,1,1,1,2,2,3,3,5,5,7,6,10,9,13,12,17,17,21,21,28,28,34,33,42,43,51,
%T A341462 53,61,63,73,76,87,91,102,104,119,123,137,143,157,164,179,187,205,215,
%U A341462 232,239,262,272,294,309,327,341,365,381,406,427,448,465
%N A341462 Number of partitions of n into 4 distinct nonprime parts.
%p A341462 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341462       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341462       `if`(isprime(i), 0, b(n-i, min(n-i, i-1), t-1))))
%p A341462     end:
%p A341462 a:= n-> b(n$2, 4):
%p A341462 seq(a(n), n=19..78);  # _Alois P. Heinz_, Feb 12 2021
%t A341462 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341462      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341462      If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1]]]];
%t A341462 a[n_] := b[n, n, 4];
%t A341462 Table[a[n], {n, 19, 78}] (* _Jean-François Alcover_, Jul 13 2021, after _Alois P. Heinz_ *)
%t A341462 Table[Length[Select[IntegerPartitions[n,{4}],Length[#]==Length[ Union[ #]] && NoneTrue[#,PrimeQ]&]],{n,19,80}] (* _Harvey P. Dale_, Nov 07 2021 *)
%Y A341462 Cf. A005171, A018252, A096258, A219198, A302479, A341451, A341461, A341464, A341465, A341466, A341467.
%K A341462 nonn
%O A341462 19,5
%A A341462 _Ilya Gutkovskiy_, Feb 12 2021