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A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.

%I A365919 #9 Sep 30 2023 09:22:35
%S A365919 1,3,9,21,22,27,63,76,81,117,147,175,186,189,243,248,273,286,290,322,
%T A365919 345,351,399,418,441,513,516,567,688,715,729,819,1029,1053,1062,1156,
%U A365919 1180,1197,1323,1375,1416,1484,1521,1539,1701,1827,1888,1911,2068,2115,2130
%N A365919 Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.
%C A365919 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F A365919 Positive integers k such that A304793(k) = A325799(k).
%e A365919 The terms together with their prime indices begin:
%e A365919      1: {}
%e A365919      3: {2}
%e A365919      9: {2,2}
%e A365919     21: {2,4}
%e A365919     22: {1,5}
%e A365919     27: {2,2,2}
%e A365919     63: {2,2,4}
%e A365919     76: {1,1,8}
%e A365919     81: {2,2,2,2}
%e A365919    117: {2,2,6}
%e A365919    147: {2,4,4}
%e A365919    175: {3,3,4}
%e A365919    186: {1,2,11}
%e A365919    189: {2,2,2,4}
%e A365919    243: {2,2,2,2,2}
%t A365919 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A365919 smu[y_]:=Union[Total/@Rest[Subsets[y]]];
%t A365919 nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t A365919 Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]
%Y A365919 The LHS is A304793, counted by A365658, with empty sets A299701.
%Y A365919 The RHS is A325799, counted by A365923 (strict A365545).
%Y A365919 A046663 counts partitions without a subset summing to k, strict A365663.
%Y A365919 A056239 adds up prime indices, row sums of A112798.
%Y A365919 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A365919 A325781 ranks complete partitions, counted by A126796.
%Y A365919 A365830 ranks incomplete partitions, counted by A365924.
%Y A365919 A365918 counts non-subset-sums of partitions, strict A365922.
%Y A365919 Cf. A001223, A005117, A006827, A073491, A188431, A304792, A365831.
%K A365919 nonn
%O A365919 1,2
%A A365919 _Gus Wiseman_, Sep 25 2023