A327670 Sum of divisors of n that have an even number of distinct prime factors.

1, 1, 1, 1, 1, 7, 1, 1, 1, 11, 1, 19, 1, 15, 16, 1, 1, 25, 1, 31, 22, 23, 1, 43, 1, 27, 1, 43, 1, 32, 1, 1, 34, 35, 36, 73, 1, 39, 40, 71, 1, 42, 1, 67, 61, 47, 1, 91, 1, 61, 52, 79, 1, 79, 56, 99, 58, 59, 1, 64, 1, 63, 85, 1, 66, 62, 1, 103, 70, 60, 1, 169, 1, 75, 91

Offset: 1

Author

Ilya Gutkovskiy, Sep 21 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000961 (positions of 1's), A000203, A030231, A049060, A285798, A318676, A327669.

Programs

Maple

with(numtheory):
a:= n-> add(`if`(nops(factorset(d))::even, d, 0), d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

Mathematica

a[n_] := DivisorSum[n, # &, EvenQ[PrimeNu[#]] &]; Table[a[n], {n, 1, 75}]

Formula

G.f.: Sum_{k>=1} A030231(k) * x^A030231(k) / (1 - x^A030231(k)).

L.g.f.: log(B(x)) = Sum_{n>=1} a(n) * x^n / n, where B(x) = g.f. of A285798.

a(n) = A000203(n) - A327669(n).