A341074 Number of partitions of n into 5 distinct squarefree parts.

1, 1, 1, 0, 2, 3, 3, 3, 5, 8, 9, 8, 11, 15, 16, 16, 22, 27, 30, 31, 38, 46, 48, 49, 57, 72, 73, 76, 90, 107, 109, 112, 128, 151, 156, 160, 182, 214, 220, 224, 250, 290, 297, 306, 335, 387, 399, 409, 442, 503, 517, 529, 572, 641, 660, 676, 726, 809, 829, 846, 903

Offset: 17

Author

Ilya Gutkovskiy, Feb 04 2021

Keywords

• nonn

Crossrefs

Cf. A005117, A008966, A098236, A307835, A308840, A341065, A341073, A341075, A341095, A341096, A341097, A341098.

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`(numtheory[issqrfree](i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 5):
seq(a(n), n=17..77);  # Alois P. Heinz, Feb 04 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[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 5];
Table[a[n], {n, 17, 77}] (* Jean-François Alcover, Jul 14 2021, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n,{5}],?(Length[Union[#]]==5&&AllTrue[#,SquareFreeQ]&)],{n,17,80}] (* _Harvey P. Dale, Sep 05 2023 *)