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

%I A341074 #11 Sep 05 2023 10:41:54
%S A341074 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,
%T A341074 76,90,107,109,112,128,151,156,160,182,214,220,224,250,290,297,306,
%U A341074 335,387,399,409,442,503,517,529,572,641,660,676,726,809,829,846,903
%N A341074 Number of partitions of n into 5 distinct squarefree parts.
%p A341074 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341074       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341074       `if`(numtheory[issqrfree](i), b(n-i, min(n-i, i-1), t-1), 0)))
%p A341074     end:
%p A341074 a:= n-> b(n$2, 5):
%p A341074 seq(a(n), n=17..77);  # _Alois P. Heinz_, Feb 04 2021
%t A341074 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341074      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341074      If[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
%t A341074 a[n_] := b[n, n, 5];
%t A341074 Table[a[n], {n, 17, 77}] (* _Jean-François Alcover_, Jul 14 2021, after _Alois P. Heinz_ *)
%t A341074 Table[Count[IntegerPartitions[n,{5}],_?(Length[Union[#]]==5&&AllTrue[#,SquareFreeQ]&)],{n,17,80}] (* _Harvey P. Dale_, Sep 05 2023 *)
%Y A341074 Cf. A005117, A008966, A098236, A307835, A308840, A341065, A341073, A341075, A341095, A341096, A341097, A341098.
%K A341074 nonn
%O A341074 17,5
%A A341074 _Ilya Gutkovskiy_, Feb 04 2021