A365830 Heinz numbers of incomplete integer partitions, meaning not every number from 0 to A056239(n) is the sum of some submultiset.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Author

Gus Wiseman, Sep 26 2023

Keywords

• nonn

Comments

First differs from A325798 in lacking 156.

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.

The complement (complete partitions) is A325781.

Examples

			The terms together with their prime indices begin:
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
For example, the submultisets of (1,1,2,6) (right column) and their sums (left column) are:
   0: ()
   1: (1)
   2: (2)  or (11)
   3: (12)
   4: (112)
   6: (6)
   7: (16)
   8: (26) or (116)
   9: (126)
  10: (1126)
But 5 is missing, so 156 is in the sequence.
		

Crossrefs

For prime indices instead of sums we have A080259, complement of A055932.

The complement is A325781, counted by A126796, strict A188431.

Positions of nonzero terms in A325799, complement A304793.

These partitions are counted by A365924, strict A365831.

A056239 adds up prime indices, row sums of A112798.

A276024 counts positive subset-sums of partitions, strict A284640

A299701 counts distinct subset-sums of prime indices.

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

A365923 counts partitions by distinct non-subset-sums, strict A365545.

Cf. A001223, A046663, A073491, A098743, A304792, A365658, A365919.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
Select[Range[100],Length[nmz[prix[#]]]>0&]