A353867 Heinz numbers of integer partitions where every partial run (consecutive constant subsequence) has a different sum, and these sums include every integer from 0 to the greatest part.

1, 2, 4, 6, 8, 16, 20, 30, 32, 56, 64, 90, 128, 140, 176, 210, 256, 416, 512, 616, 990, 1024, 1088, 1540, 2048, 2288, 2310, 2432, 2970, 4096, 4950, 5888, 7072, 7700, 8008, 8192, 11550, 12870, 14848, 16384, 20020, 20672, 30030, 31744, 32768, 38896, 50490, 55936

Offset: 1

Author

Gus Wiseman, Jun 07 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.

Related concepts:

- A partition whose submultiset sums cover an initial interval is said to be complete (A126796, ranked by A325781).

- In a knapsack partition (A108917, ranked by A299702), every submultiset has a different sum.

- A complete partition that is also knapsack is said to be perfect (A002033, ranked by A325780).

- A partition whose partial runs have all different sums is said to be rucksack (A353864, ranked by A353866, complement A354583).

Examples

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   16: {1,1,1,1}
   20: {1,1,3}
   30: {1,2,3}
   32: {1,1,1,1,1}
   56: {1,1,1,4}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  140: {1,1,3,4}
  176: {1,1,1,1,5}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}
		

Crossrefs

Knapsack partitions are counted by A108917, ranked by A299702.

Complete partitions are counted by A126796, ranked by A325781.

These partitions are counted by A353865.

This is a special case of A353866, counted by A353864, complement A354583.

A001222 counts prime factors, distinct A001221.

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

A073093 counts prime-power divisors.

A124010 gives prime signature, sorted A118914.

A300273 ranks collapsible partitions, counted by A275870.

A353832 represents the operation of taking run-sums of a partition.

A353833 ranks partitions with all equal run-sums, nonprime A353834.

A353836 counts partitions by number of distinct run-sums.

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

A353863 counts partitions whose weak run-sums cover an initial interval.

Cf. A018818, A181819, A182857, A304442, A316413, A325862, A353835, A353838, A353839, A353861, A353931.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
norqQ[m_]:=Sort[m]==Range[0,Max[m]];
msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
Select[Range[1000],norqQ[Total/@Select[msubs[primeMS[#]],SameQ@@#&]]&]