cp's OEIS Frontend

A381959 Denominator of the sum of the reciprocals of the indices of distinct prime factors of n.

%I A381959 #11 Mar 19 2025 15:30:21
%S A381959 1,1,2,1,3,2,4,1,2,3,5,2,6,4,6,1,7,2,8,3,4,5,9,2,3,6,2,4,10,6,11,1,10,
%T A381959 7,12,2,12,8,3,3,13,4,14,5,6,9,15,2,4,3,14,6,16,2,15,4,8,10,17,6,18,
%U A381959 11,4,1,2,10,19,7,18,12,20,2,21,12,6,8,20,3,22,3,2,13,23,4,21,14,5,5,24,6,12,9,22,15,24
%N A381959 Denominator of the sum of the reciprocals of the indices of distinct prime factors of n.
%H A381959 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F A381959 If n = Product (p_j^k_j) then a(n) = denominator of Sum (1/pi(p_j)).
%F A381959 G.f. for fractions: Sum_{k>=1} x^prime(k) / (k*(1 - x^prime(k))).
%e A381959 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 A381959 Join[{1}, Table[Plus @@ (1/PrimePi[#[[1]]] & /@ FactorInteger[n]), {n, 2, 95}] // Denominator]
%t A381959 nmax = 95; CoefficientList[Series[Sum[x^Prime[k]/(k (1 - x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest
%o A381959 (PARI) a(n) = my(f=factor(n)); denominator(sum(k=1, #f~, 1/primepi(f[k,1]))); \\ _Michel Marcus_, Mar 11 2025
%Y A381959 Cf. A000720, A007947, A066328, A083346, A318574, A381958 (numerators).
%K A381959 nonn,frac
%O A381959 1,3
%A A381959 _Ilya Gutkovskiy_, Mar 11 2025