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

1, 3, 9, 21, 22, 27, 63, 76, 81, 117, 147, 175, 186, 189, 243, 248, 273, 286, 290, 322, 345, 351, 399, 418, 441, 513, 516, 567, 688, 715, 729, 819, 1029, 1053, 1062, 1156, 1180, 1197, 1323, 1375, 1416, 1484, 1521, 1539, 1701, 1827, 1888, 1911, 2068, 2115, 2130

Offset: 1

Author

Gus Wiseman, Sep 25 2023

Keywords

• nonn

Comments

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.

Examples

			The terms together with their prime indices begin:
     1: {}
     3: {2}
     9: {2,2}
    21: {2,4}
    22: {1,5}
    27: {2,2,2}
    63: {2,2,4}
    76: {1,1,8}
    81: {2,2,2,2}
   117: {2,2,6}
   147: {2,4,4}
   175: {3,3,4}
   186: {1,2,11}
   189: {2,2,2,4}
   243: {2,2,2,2,2}
		

Crossrefs

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

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

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

A056239 adds up prime indices, row sums of A112798.

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

A325781 ranks complete partitions, counted by A126796.

A365830 ranks incomplete partitions, counted by A365924.

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

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

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
smu[y_]:=Union[Total/@Rest[Subsets[y]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[smu[prix[#]]]==Length[nmz[prix[#]]]&]

Formula

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