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

1,1

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).

Links

Table of n, a(n) for n=1..54.

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

Example

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}

Mathematica

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

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.

Sequence in context: A102749 A085231 A057715 * A053990 A026365 A051435

Adjacent sequences: A357860 A357861 A357862 * A357864 A357865 A357866

Keyword

nonn

Author

Gus Wiseman, Oct 19 2022

Status

approved