A357139 Take the weakly increasing prime indices of each prime index of n, then concatenate.

1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 1, 2, 5, 1, 3, 4, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 6, 1, 1, 1, 1, 4, 3, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 1, 1, 2, 2, 1, 4, 1, 2

Offset: 1

Author

Gus Wiseman, Sep 29 2022

Keywords

• nonn
• tabf

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.

Examples

			Triangle begins:
   1:
   2:
   3:  1
   4:
   5:  2
   6:  1
   7:  1 1
   8:
   9:  1 1
  10:  2
  11:  3
  12:  1
  13:  1 2
For example, the weakly increasing prime indices of 105 are (2,3,4), with prime indices ((1),(2),(1,1)), so row 105 is (1,2,1,1).
		

Crossrefs

Row lengths are A302242.

Positions of strict rows are A302505.

Positions of constant rows are A302593.

Row sums are A325033, products A325032.

The version for standard compositions is A357135, rank A357134.

A000961 lists prime powers.

A003963 multiples prime indices.

A056239 adds up prime indices.

Cf. A000720, A001221, A001222, A007716, A058891, A109082, A275024, A302243, A324926, A325034.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@Table[Join@@primeMS/@primeMS[n],{n,100}]