A366784 Sum of even indices of distinct prime factors of n divided by 2.

0, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 4, 0, 3, 0, 0, 1, 0, 3, 1, 2, 5, 1, 0, 0, 1, 0, 2, 1, 6, 4, 4, 0, 0, 3, 7, 0, 1, 0, 0, 1, 2, 0, 1, 3, 8, 1, 0, 2, 5, 5, 0, 1, 9, 0, 3, 0, 3, 1, 0, 0, 1, 2, 10, 1, 0, 6, 1, 4, 2, 4, 11, 0, 1, 0, 0, 3, 0, 7, 6, 0, 12, 1, 5, 0, 1, 0, 4, 1, 0, 2, 1

Offset

1,7

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

G.f.: Sum_{k>=1} k * x^prime(2*k) / (1 - x^prime(2*k)).

From Amiram Eldar, Jul 03 2025: (Start)

Additive with a(p^e) = pi(p)/2 if pi(p) is even, and 0 otherwise.

a(n) = (A066328(n) - A366725(n))/2. (End)

Example

a(315) = 3 because 315 = 3^2 * 5 * 7 = prime(2)^2 * prime(3) * prime(4) and (2 + 4) / 2 = 3.

Mathematica

nmax = 100; CoefficientList[Series[Sum[k x^Prime[2 k]/(1 - x^Prime[2 k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := Module[{i = PrimePi[p]}, If[EvenQ[i], i/2, 0]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 03 2025 *)

Prog

(PARI) f(n) = if(n % 2, 0, n/2);
a(n) = vecsum(apply(x -> f(primepi(x)), factor(n)[, 1])); \\ Amiram Eldar, Jul 03 2025

Crossrefs

Cf. A000720 (pi), A066208 (positions of 0's), A066328, A324967, A332422, A344931, A366533, A366725.

Sequence in context: A144628 A373832 A286604 * A217540 A226861 A185643

Adjacent sequences: A366781 A366782 A366783 * A366785 A366786 A366787

Keyword

nonn

Author

Ilya Gutkovskiy, Oct 24 2023

Status

approved