A357863 Numbers whose prime indices do not have strictly increasing run-sums. Heinz numbers of the partitions not counted by A304428.

12, 24, 40, 45, 48, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 180, 189, 192, 204, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405, 408, 420, 440

Offset: 1

Author

Gus Wiseman, Oct 19 2022

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

Examples

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   40: {1,1,1,3}
   45: {2,2,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  135: {2,2,2,3}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
		

Links

• Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Crossrefs

These are the indices of rows in A354584 that are not strictly increasing.

The complement (strictly increasing) is A357862, counted by A304428.

The weak (not weakly increasing) version is A357876, counted by A357878.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A118914, A181819, A300273, A304430, A304442, A357864, A357875.

Programs

Mathematica

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