A332422 If n = Product (p_j^k_j) then a(n) = Sum ((-1)^(pi(p_j) + 1) * pi(p_j)), where pi = A000720.

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

Offset

1,3

Comments

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

Links

Table of n, a(n) for n=1..72.

Index entries for sequences computed from indices in prime factorization

Formula

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

Example

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

Mathematica

a[n_] := Plus @@ ((-1)^(PrimePi[#[[1]]] + 1) PrimePi[#[[1]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 72}]
nmax = 72; CoefficientList[Series[Sum[(-1)^(k + 1) k x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Crossrefs

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

Sequence in context: A347103 A255810 A210256 * A229944 A366421 A218533

Adjacent sequences: A332419 A332420 A332421 * A332423 A332424 A332425

Keyword

sign

Author

Ilya Gutkovskiy, Feb 12 2020

Status

approved