A357876 The run-sums of the prime indices of n are not weakly increasing.

24, 45, 48, 80, 90, 96, 120, 135, 160, 168, 175, 180, 189, 192, 224, 240, 264, 270, 275, 288, 297, 312, 315, 320, 336, 350, 360, 378, 384, 405, 408, 448, 456, 480, 495, 525, 528, 539, 540, 550, 552, 560, 567, 576, 585, 594, 600, 624, 630, 637, 640, 672, 696

Offset: 1

Author

Gus Wiseman, Oct 17 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 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:
   24: {1,1,1,2}
   45: {2,2,3}
   48: {1,1,1,1,2}
   80: {1,1,1,1,3}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  135: {2,2,2,3}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
  175: {3,3,4}
  180: {1,1,2,2,3}
  189: {2,2,2,4}
  192: {1,1,1,1,1,1,2}
For example, the prime indices of 24 are (1,1,1,2), with run-sums (3,2), which are not weakly increasing, so 24 is in the sequence.
		

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 weakly increasing.

The complement is A357875.

These partitions are counted by A357878.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A047966, A118914, A181819, A239312, A275870, A300273, A304442, A325249, A353743-A354912.

Programs

Mathematica

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