A357861 Numbers whose prime indices have weakly decreasing run-sums. Heinz numbers of the partitions counted by A304406.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 40, 41, 43, 45, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 135, 137, 139, 144, 149, 151, 157

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:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are weakly decreasing, so 24 is in the sequence.
		

Links

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

Crossrefs

These partitions are counted by A304406.

These are the indices of rows in A354584 that are weakly decreasing.

The complement is A357850, counted by A357865, opposite A357876.

The strictly decreasing version is A357864, counted by A304430.

The opposite (weakly increasing) version is A357875, counted by A304405.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

Cf. A047966, A118914, A181819, A239312, A300273, A304442.

Programs

Mathematica

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