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%I #15 Nov 25 2023 23:54:37
%S 1,2,2,3,2,4,2,4,3,4,2,5,2,4,4,5,2,6,2,6,4,4,2,6,3,4,4,6,2,7,2,6,4,4,
%T 4,7,2,4,4,7,2,8,2,6,6,4,2,7,3,6,4,6,2,8,4,8,4,4,2,8,2,4,5,7,4,8,2,6,
%U 4,7,2,8,2,4,6,6,4,8,2,8,5,4,2,9,4,4,4
%N Number of distinct subset-sums of the integer partition with Heinz number n.
%C An integer n is a 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 Position of first appearance of n appears to be A259941(n) = least Heinz number of a complete partition of n. - _Gus Wiseman_, Nov 16 2023
%H Alois P. Heinz, <a href="/A299701/b299701.txt">Table of n, a(n) for n = 1..20000</a>
%F a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).
%e The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
%e The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.
%t Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]
%Y Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
%Y Positions of first appearances are A259941.
%Y The triangle for this rank statistic is A365658.
%Y The semi version is A366739, sum A366738, strict A366741.
%Y Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.
%K nonn
%O 1,2
%A _Gus Wiseman_, Feb 17 2018
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