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%I #7 Aug 03 2023 09:04:19
%S 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,25,26,27,28,
%T 29,31,32,33,34,35,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,54,55,
%U 56,57,58,59,61,62,64,65,66,67,68,69,71,73,74,75,76,77
%N Positive integers with no prime index equal to the sum of prime indices of any nonprime divisor.
%C First differs from A299702 (knapsack) in having 525: {2,3,3,4}.
%C First differs from A325778 in lacking 462: {1,2,4,5}.
%C These are the Heinz numbers of partitions whose parts are disjoint from their own non-singleton subset-sums.
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]=={}&]
%Y Partitions of this type are counted by A237667, strict A364349.
%Y The binary version is A364462, complement A364461.
%Y The complement is A364532, counted by A237668.
%Y A000005 counts divisors, nonprime A033273, composite A055212.
%Y A299701 counts distinct subset-sums of prime indices.
%Y A299702 ranks knapsack partitions, counted by A108917, complement A299729.
%Y A363260 counts partitions disjoint from differences, complement A364467.
%Y Cf. A002865, A236912, A237113, A320340, A326083, A363225, A364347.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 01 2023