A332422 - OEIS

%I #5 Feb 13 2020 02:53:45

%S 0,1,-2,1,3,-1,-4,1,-2,4,5,-1,-6,-3,1,1,7,-1,-8,4,-6,6,9,-1,3,-5,-2,

%T -3,-10,2,11,1,3,8,-1,-1,-12,-7,-8,4,13,-5,-14,6,1,10,15,-1,-4,4,5,-5,

%U -16,-1,8,-3,-10,-9,17,2,-18,12,-6,1,-3,4,19,8,7,0,-20,-1

%N If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(pi(p_j) + 1) * pi(p_j)), where pi = A000720.

%C Sum of odd indices of distinct prime factors of n minus the sum of even indices of distinct prime factors of n.

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F G.f.: Sum_{k>=1} (-1)^(k + 1) * k * x^prime(k) / (1 - x^prime(k)).

%e a(66) = a(2 * 3 * 11) = a(prime(1) * prime(2) * prime(5)) = 1 - 2 + 5 = 4.

%t a[n_] := Plus @@ ((-1)^(PrimePi[#[[1]]] + 1) PrimePi[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 72}]

%t nmax = 72; CoefficientList[Series[Sum[(-1)^(k + 1) k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%Y Cf. A000720, A002129, A071321, A112798, A066328, A195017, A316524, A325699, A332423.

%K sign

%O 1,3

%A _Ilya Gutkovskiy_, Feb 12 2020