A321144 Irregular triangle where T(n,k) is the number of divisors of n whose prime indices sum to k.

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Author

Gus Wiseman, Oct 28 2018

Keywords

• nonn
• tabf

Comments

The rows are all palindromes.

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
  1  1
  1  0  1
  1  1  1
  1  0  0  1
  1  1  1  1
  1  0  0  0  1
  1  1  1  1
  1  0  1  0  1
  1  1  0  1  1
  1  0  0  0  0  1
  1  1  2  1  1
  1  0  0  0  0  0  1
  1  1  0  0  1  1
  1  0  1  1  0  1
  1  1  1  1  1
  1  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  0  0  0  0  0  0  1
  1  1  1  1  1  1
  1  0  1  0  1  0  1
  1  1  0  0  0  1  1
  1  0  0  0  0  0  0  0  0  1
  1  1  2  2  1  1
  1  0  0  1  0  0  1
  1  1  0  0  0  0  1  1
  1  0  1  0  1  0  1
  1  1  1  0  1  1  1
  1  0  0  0  0  0  0  0  0  0  1
  1  1  1  2  1  1  1
		

Crossrefs

Row lengths are A056239. Number of nonzero entries in row n is A299701(n).

Cf. A000005, A112798, A276024, A301854, A301899.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
Table[Count[Total/@primeMS/@Divisors[n],k],{n,20},{k,0,Total[primeMS[n]]}]