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

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, 53, 61, 63, 73, 76, 87, 91, 102, 104, 119, 123, 137, 143, 157, 164, 179, 187, 205, 215, 232, 239, 262, 272, 294, 309, 327, 341, 365, 381, 406, 427, 448, 465

Offset: 19

Author

Ilya Gutkovskiy, Feb 12 2021

Keywords

• nonn

Crossrefs

Cf. A005171, A018252, A096258, A219198, A302479, A341451, A341461, A341464, A341465, A341466, A341467.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(isprime(i), 0, b(n-i, min(n-i, i-1), t-1))))
    end:
a:= n-> b(n$2, 4):
seq(a(n), n=19..78);  # Alois P. Heinz, Feb 12 2021

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[PrimeQ[i], 0, b[n - i, Min[n - i, i - 1], t - 1]]]];
a[n_] := b[n, n, 4];
Table[a[n], {n, 19, 78}] (* Jean-François Alcover, Jul 13 2021, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n,{4}],Length[#]==Length[ Union[ #]] && NoneTrue[#,PrimeQ]&]],{n,19,80}] (* Harvey P. Dale, Nov 07 2021 *)