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A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

%I A304793 #14 Jan 27 2019 02:57:21
%S A304793 0,1,1,2,1,3,1,3,2,3,1,4,1,3,3,4,1,5,1,5,3,3,1,5,2,3,3,5,1,6,1,5,3,3,
%T A304793 3,6,1,3,3,6,1,7,1,5,5,3,1,6,2,5,3,5,1,7,3,7,3,3,1,7,1,3,4,6,3,7,1,5,
%U A304793 3,6,1,7,1,3,5,5,3,7,1,7,4,3,1,8,3,3,3,7,1,8,3,5,3,3,3,7,1,5,5,8,1,7,1,7,7
%N A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.
%C A304793 A positive integer n is a positive subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A304793 a(n) <= A000005(n).
%C A304793 One less than the number of distinct values obtained when A056239 is applied to all divisors of n. - _Antti Karttunen_, Jul 01 2018
%H A304793 Antti Karttunen, <a href="/A304793/b304793.txt">Table of n, a(n) for n = 1..65537</a>
%e A304793 The positive subset-sums of (4,3,1) are {1, 3, 4, 5, 7, 8} so a(70) = 6.
%e A304793 The positive subset-sums of (5,1,1,1) are {1, 2, 3, 5, 6, 7, 8} so a(88) = 7.
%t A304793 Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]
%o A304793 (PARI)
%o A304793 up_to = 65537;
%o A304793 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
%o A304793 v056239 = vector(up_to,n,A056239(n));
%o A304793 A304793(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v056239[d]), mapput(m,s,s); k++)); (k-1); }; \\ _Antti Karttunen_, Jul 01 2018
%Y A304793 Cf. A056239, A122768, A276024, A284640, A296150, A299701, A299702, A301855, A301935, A301957, A304792, A304795, A305611.
%K A304793 nonn
%O A304793 1,4
%A A304793 _Gus Wiseman_, May 18 2018
%E A304793 More terms from _Antti Karttunen_, Jul 01 2018