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A366725 Sum of odd indices of distinct prime factors of n.

%I A366725 #24 Jul 03 2025 01:02:26
%S A366725 0,1,0,1,3,1,0,1,0,4,5,1,0,1,3,1,7,1,0,4,0,6,9,1,3,1,0,1,0,4,11,1,5,8,
%T A366725 3,1,0,1,0,4,13,1,0,6,3,10,15,1,0,4,7,1,0,1,8,1,0,1,17,4,0,12,0,1,3,6,
%U A366725 19,8,9,4,0,1,21,1,3,1,5,1,0,4,0,14,23,1,10,1,0,6,0,4,0,10,11,16,3,1,25,1,5,4
%N A366725 Sum of odd indices of distinct prime factors of n.
%H A366725 Amiram Eldar, <a href="/A366725/b366725.txt">Table of n, a(n) for n = 1..10000</a>
%F A366725 G.f.: Sum_{k>=1} (2*k-1) * x^prime(2*k-1) / (1 - x^prime(2*k-1)).
%F A366725 From _Amiram Eldar_, Jul 03 2025: (Start)
%F A366725 Additive with a(p^e) = pi(p) if pi(p) is odd, and 0 otherwise.
%F A366725 a(n) = A066328(n) - 2*A366784(n). (End)
%e A366725 a(60) = 4 because 60 = 2^2 * 3 * 5 = prime(1)^2 * prime(2) * prime(3) and 1 + 3 = 4.
%t A366725 nmax = 100; CoefficientList[Series[Sum[(2 k - 1) x^Prime[2 k - 1]/(1 - x^Prime[2 k - 1]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
%t A366725 f[p_, e_] := Module[{i = PrimePi[p]}, If[OddQ[i], i, 0]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 03 2025 *)
%o A366725 (PARI) f(n) = if(n % 2, n, 0);
%o A366725 a(n) = vecsum(apply(x -> f(primepi(x)), factor(n)[, 1])); \\ _Amiram Eldar_, Jul 03 2025
%Y A366725 Cf. A000720 (pi), A066207 (positions of 0's), A066328, A324966, A332422, A344908, A366528, A366784.
%K A366725 nonn
%O A366725 1,5
%A A366725 _Ilya Gutkovskiy_, Oct 24 2023