cp's OEIS Frontend

A381958 Numerator of the sum of the reciprocals of the indices of distinct prime factors of n.

%I A381958 #12 Mar 19 2025 15:30:15
%S A381958 0,1,1,1,1,3,1,1,1,4,1,3,1,5,5,1,1,3,1,4,3,6,1,3,1,7,1,5,1,11,1,1,7,8,
%T A381958 7,3,1,9,2,4,1,7,1,6,5,10,1,3,1,4,9,7,1,3,8,5,5,11,1,11,1,12,3,1,1,17,
%U A381958 1,8,11,19,1,3,1,13,5,9,9,5,1,4,1,14,1,7,10,15,3,6,1,11,5,10,13,16,11
%N A381958 Numerator of the sum of the reciprocals of the indices of distinct prime factors of n.
%H A381958 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A381958 If n = Product (p_j^k_j) then a(n) = numerator of Sum (1/pi(p_j)).
%F A381958 G.f. for fractions: Sum_{k>=1} x^prime(k) / (k*(1 - x^prime(k))).
%e A381958 0, 1, 1/2, 1, 1/3, 3/2, 1/4, 1, 1/2, 4/3, 1/5, 3/2, 1/6, 5/4, 5/6, 1, 1/7, 3/2, 1/8, 4/3, ...
%t A381958 Join[{0}, Table[Plus @@ (1/PrimePi[#[[1]]] & /@ FactorInteger[n]), {n, 2, 95}] // Numerator]
%t A381958 nmax = 95; CoefficientList[Series[Sum[x^Prime[k]/(k (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest
%o A381958 (PARI) a(n) = my(f=factor(n)); numerator(sum(k=1, #f~, 1/primepi(f[k,1]))); \\ _Michel Marcus_, Mar 11 2025
%Y A381958 Cf. A000720, A028235, A028236, A066328, A083345, A318573, A379141, A381959 (denominators).
%K A381958 nonn,frac
%O A381958 1,6
%A A381958 _Ilya Gutkovskiy_, Mar 11 2025