A341453 Number of partitions of n into 6 nonprime parts.

1, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 10, 12, 15, 16, 20, 23, 27, 30, 36, 40, 48, 53, 62, 68, 81, 87, 105, 112, 130, 141, 166, 176, 208, 219, 256, 271, 314, 331, 385, 403, 468, 488, 561, 588, 674, 702, 804, 837, 952, 991, 1126, 1168, 1321, 1372

Offset: 6

Author

Ilya Gutkovskiy, Feb 12 2021

Keywords

• nonn

Crossrefs

Cf. A002095, A005171, A018252, A062610, A259196, A341408, A341451, A341452, A341454, A341455, A341457.

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), t-1))))
    end:
a:= n-> b(n$2, 6):
seq(a(n), n=6..67);  # 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], t - 1], 0]]];
a[n_] := b[n, n, 6];
Table[a[n], {n, 6, 67}] (* Jean-François Alcover, Feb 23 2022, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{6}],?(NoneTrue[#,PrimeQ]&)],{n,6,70}] (* _Harvey P. Dale, Feb 21 2023 *)