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

1,45

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.

Links

Table of n, a(n) for n=1..87.

Example

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

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]]}]

Crossrefs

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

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

Sequence in context: A377080 A316867 A127327 * A309048 A379666 A086072

Adjacent sequences: A321141 A321142 A321143 * A321145 A321146 A321147

Keyword

nonn,tabf

Author

Gus Wiseman, Oct 28 2018

Status

approved