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

1,4

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

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A000203, A001221.

Sequence in context: A331147 A208508 A123726 * A336851 A138675 A274091

Adjacent sequences: A323597 A323598 A323599 * A323601 A323602 A323603

Keyword

nonn

Author

Torlach Rush, Jan 18 2019

Status

approved