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Heinz numbers of integer partitions with the same number of distinct positive subset-sums as distinct non-subset-sums.
6

%I #9 Sep 30 2023 09:22:35

%S 1,3,9,21,22,27,63,76,81,117,147,175,186,189,243,248,273,286,290,322,

%T 345,351,399,418,441,513,516,567,688,715,729,819,1029,1053,1062,1156,

%U 1180,1197,1323,1375,1416,1484,1521,1539,1701,1827,1888,1911,2068,2115,2130

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

%C 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 Positive integers k such that A304793(k) = A325799(k).

%e The terms together with their prime indices begin:

%e 1: {}

%e 3: {2}

%e 9: {2,2}

%e 21: {2,4}

%e 22: {1,5}

%e 27: {2,2,2}

%e 63: {2,2,4}

%e 76: {1,1,8}

%e 81: {2,2,2,2}

%e 117: {2,2,6}

%e 147: {2,4,4}

%e 175: {3,3,4}

%e 186: {1,2,11}

%e 189: {2,2,2,4}

%e 243: {2,2,2,2,2}

%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t smu[y_]:=Union[Total/@Rest[Subsets[y]]];

%t nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];

%t Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]

%Y The LHS is A304793, counted by A365658, with empty sets A299701.

%Y The RHS is A325799, counted by A365923 (strict A365545).

%Y A046663 counts partitions without a subset summing to k, strict A365663.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A276024 counts positive subset-sums of partitions, strict A284640.

%Y A325781 ranks complete partitions, counted by A126796.

%Y A365830 ranks incomplete partitions, counted by A365924.

%Y A365918 counts non-subset-sums of partitions, strict A365922.

%Y Cf. A001223, A005117, A006827, A073491, A188431, A304792, A365831.

%K nonn

%O 1,2

%A _Gus Wiseman_, Sep 25 2023