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A341142 Number of partitions of n into 5 distinct prime powers (including 1).

%I A341142 #14 Dec 22 2024 11:10:58
%S A341142 1,0,1,2,3,3,5,5,7,8,10,12,15,15,18,21,23,26,31,33,36,41,43,48,52,58,
%T A341142 62,72,72,82,85,95,97,112,112,125,127,142,142,161,159,181,180,200,196,
%U A341142 222,217,243,239,269,261,291,284,316,308,341,332,370,358,394,381,427,414,456
%N A341142 Number of partitions of n into 5 distinct prime powers (including 1).
%H A341142 Alois P. Heinz, <a href="/A341142/b341142.txt">Table of n, a(n) for n = 15..5000</a>
%p A341142 q:= proc(n) option remember; nops(ifactors(n)[2])<2 end:
%p A341142 b:= proc(n, i, t) option remember; `if`(n=0,
%p A341142       `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
%p A341142       `if`(q(i), b(n-i, min(n-i, i-1), t-1), 0)))
%p A341142     end:
%p A341142 a:= n-> b(n$2, 5):
%p A341142 seq(a(n), n=15..78);  # _Alois P. Heinz_, Feb 05 2021
%t A341142 q[n_] := q[n] = PrimeNu[n] < 2;
%t A341142 b[n_, i_, t_] := b[n, i, t] = If[n == 0,
%t A341142      If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
%t A341142      If[q[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
%t A341142 a[n_] := b[n, n, 5];
%t A341142 Table[a[n], {n, 15, 78}] (* _Jean-François Alcover_, Jul 13 2021, after _Alois P. Heinz_ *)
%t A341142 Table[Count[IntegerPartitions[n,{5}],_?(Max[PrimeNu[#]]<2&&Length[#]==Length[Union[#]]&)],{n,15,80}] (* _Harvey P. Dale_, Dec 22 2024 *)
%Y A341142 Cf. A000961, A010055, A341123, A341132, A341134, A341140, A341141, A341143, A341144, A341145.
%K A341142 nonn
%O A341142 15,4
%A A341142 _Ilya Gutkovskiy_, Feb 05 2021