A358172 Triangle read by rows: if n has weakly increasing prime indices (a,b,...,y,z) then row n is (z-a+1, z-b+1, ..., z-y+1).

1, 2, 1, 1, 1, 3, 2, 2, 4, 2, 1, 1, 1, 2, 1, 3, 3, 3, 5, 2, 2, 2, 1, 6, 1, 1, 4, 4, 3, 2, 1, 1, 1, 1, 4, 7, 2, 2, 2, 1, 8, 5, 3, 3, 3, 4, 3, 5, 5, 2, 2, 9, 2, 2, 2, 2, 1, 3, 1, 6, 6, 6, 2, 1, 1, 3, 4, 4, 4, 7, 10, 3, 3, 2, 11, 3, 3, 1, 1, 1, 1, 1, 4, 5, 4

Offset: 1

Author

Gus Wiseman, Dec 20 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:   .
   4:   1
   5:   .
   6:   2
   7:   .
   8:  1 1
   9:   1
  10:   3
  11:   .
  12:  2 2
  13:   .
  14:   4
  15:   2
  16: 1 1 1
  17:   .
  18:  2 1
  19:   .
  20:  3 3
For example, the prime indices of 900 are (1,1,2,2,3,3), so row 900 is 3 - (1,1,2,2,3) + 1 = (3,3,2,2,1).
		

Crossrefs

Row lengths are A001222(n) - 1.

Indices of empty rows are A008578.

Even-indexed rows have sums A243503.

Row sums are A326844(n) + A001222(n) - 1.

An opposite version is A356958, Heinz numbers A246277.

Heinz numbers of the rows are A358195, even bisection A241916.

A112798 list prime indices, sum A056239.

A243055 subtracts the least prime index from the greatest.

Cf. A055396, A124010, A253565, A325351, A325352, A355534, A355536, A358137.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,{},1+Last[primeMS[n]]-Most[primeMS[n]]],{n,100}]