cp's OEIS Frontend

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

%I A367095 #14 Jan 20 2025 10:41:12
%S A367095 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,
%T A367095 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,
%U A367095 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
%N A367095 Number of distinct sums of pairs (repeats allowed) of prime indices of n.
%C A367095 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.
%C A367095 Is the image missing only 2 and 4?
%H A367095 Antti Karttunen, <a href="/A367095/b367095.txt">Table of n, a(n) for n = 1..65537</a>
%H A367095 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%e A367095 The prime indices of 15 are {2,3}, with sums of pairs:
%e A367095   2+2 = 4
%e A367095   2+3 = 5
%e A367095   3+3 = 6
%e A367095 so a(15) = 3.
%e A367095 The prime indices of 180 are {1,1,2,2,3}, with sums of pairs:
%e A367095   1+1 = 2
%e A367095   1+2 = 3
%e A367095   1+3 = 4
%e A367095   2+2 = 4
%e A367095   2+3 = 5
%e A367095   3+3 = 6
%e A367095 so a(180) = 5.
%t A367095 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A367095 Table[Length[Union[Total/@Tuples[prix[n],2]]],{n,100}]
%o A367095 (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
%Y A367095 Depends only on squarefree kernel A007947. (Even more exactly, on A322591 - _Antti Karttunen_, Jan 20 2025)
%Y A367095 Positions of first appearances appear to be a subset of A325986.
%Y A367095 For 2-element submultisets we have A366739, for all submultisets A299701.
%Y A367095 A001222 counts prime factors (also indices), distinct A001221.
%Y A367095 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A367095 A056239 adds up prime indices, row sums of A112798.
%Y A367095 A304793 counts positive subset-sums of prime indices.
%Y A367095 A367096 lists semiprime divisors, count A086971.
%Y A367095 Cf. A001248, A004526, A008967, A366737, A366738, A366740, A367093, A367097.
%K A367095 nonn
%O A367095 1,6
%A A367095 _Gus Wiseman_, Nov 06 2023
%E A367095 Data section extended to a(105) by _Antti Karttunen_, Jan 20 2025