A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 3, 1, 1, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 5, 1, 1, 3, 3, 3, 3, 1, 3, 3, 3, 1, 6, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 3, 3, 3, 3, 3, 1, 5, 1, 3, 3, 1, 3, 6, 1, 3, 3, 6, 1, 3, 1, 3, 3, 3, 3, 6, 1, 3, 1, 3, 1, 6, 3, 3, 3

Offset

1,6

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.

Is the image missing only 2 and 4?

Links

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

Example

The prime indices of 15 are {2,3}, with sums of pairs:
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(15) = 3.
The prime indices of 180 are {1,1,2,2,3}, with sums of pairs:
  1+1 = 2
  1+2 = 3
  1+3 = 4
  2+2 = 4
  2+3 = 5
  3+3 = 6
so a(180) = 5.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Union[Total/@Tuples[prix[n], 2]]], {n, 100}]

Crossrefs

Depends only on squarefree kernel A007947.

Positions of first appearances appear to be a subset of A325986.

For 2-element submultisets we have A366739, for all submultisets A299701.

A001222 counts prime factors (also indices), distinct A001221.

A001358 lists semiprimes, squarefree A006881, conjugate A065119.

A056239 adds up prime indices, row sums of A112798.

A304793 counts positive subset-sums of prime indices.

A367096 lists semiprime divisors, count A086971.

Cf. A001248, A004526, A008967, A366737, A366738, A366740, A367093, A367097.

Sequence in context: A353235 A243915 A367482 * A309307 A325446 A337322

Adjacent sequences: A367092 A367093 A367094 * A367096 A367097 A367098

Keyword

nonn

Author

Gus Wiseman, Nov 06 2023

Status

approved