A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

12, 24, 36, 40, 48, 60, 63, 72, 80, 84, 96, 108, 112, 120, 126, 132, 144, 156, 160, 168, 180, 189, 192, 200, 204, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 320, 324, 325, 336, 348, 351, 352, 360, 372, 378, 384, 396, 400, 408, 420, 432, 440

Offset: 1

Author

Gus Wiseman, Jun 15 2022

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.

The term rucksack is short for run-knapsack.

Examples

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.
		

Crossrefs

Knapsack partitions are counted by A108917, ranked by A299702.

Non-knapsack partitions are ranked by A299729.

The non-partial version is A353839, complement A353838 (counted by A353837).

The complement is A353866, counted by A353864.

The complete complement is A353867, counted by A353865.

The complement for compositions is counted by A354580.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A073093 counts prime-power divisors.

A300273 ranks collapsible partitions, counted by A275870.

A304442 counts partitions with all equal run-sums, ranked by A353833.

A333223 ranks knapsack compositions, counted by A325676.

A353852 ranks compositions with all distinct run-sums, counted by A353850.

A353861 counts distinct partial run-sums of prime indices.

A354584 lists run-sums of prime indices, rows ranked by A353832.

Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, A353931.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@Total/@primeMS/@Select[Divisors[#],PrimePowerQ]&]