A323600 Dirichlet convolution of sigma with omega.

0, 1, 1, 4, 1, 9, 1, 11, 5, 11, 1, 31, 1, 13, 12, 26, 1, 38, 1, 39, 14, 17, 1, 81, 7, 19, 18, 47, 1, 83, 1, 57, 18, 23, 16, 127, 1, 25, 20, 103, 1, 101, 1, 63, 53, 29, 1, 187, 9, 66, 24, 71, 1, 130, 20, 125, 26, 35, 1, 272, 1, 37, 63, 120, 22, 137, 1, 87, 30, 127, 1

Offset: 1

Author

Torlach Rush, Jan 18 2019

Keywords

• nonn

Comments

a(n) = omega(n) = 1 iff n is prime.

Not all positive integers are terms of this sequence as many are not expressible as the sum of products defined by the sequence, for example 2, 3, and 6.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Wikipedia, Dirichlet convolution

Crossrefs

Cf. A000203, A001221.

Programs

Maple

with(numtheory):
a:= n-> add(sigma(d)*nops(factorset(n/d)), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 28 2019

Mathematica

Table[DivisorSum[n, DivisorSigma[1, #] PrimeNu[n/#] &], {n, 71}] (* Michael De Vlieger, Jan 27 2019 *)

PARI

a(n) = sumdiv(n, d, sigma(d)*omega(n/d)); \\ Michel Marcus, Jan 22 2019

Formula

a(n) = Sum_{d|n} A000203(d) * A001221(n/d).