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

Offset: 1

Author

Gus Wiseman, Nov 06 2023

Keywords

• nonn

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?

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to prime indices in the factorization of n.

Crossrefs

Depends only on squarefree kernel A007947. (Even more exactly, on A322591 - Antti Karttunen, Jan 20 2025)

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.

Programs

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}]

PARI

A367095(n) = if(1==n, 0, my(pis=apply(primepi,factor(n)[,1]), pairsums = vector(binomial(1+#pis,2)), k=0); for(i=1,#pis,for(j=i,#pis,k++; pairsums[k] = pis[i]+pis[j])); #Set(pairsums)); \\ Antti Karttunen, Jan 20 2025

Extensions

Data section extended to a(105) by Antti Karttunen, Jan 20 2025