A361706 Inverse Moebius transform applied twice to primes.

2, 7, 9, 19, 15, 37, 21, 50, 39, 65, 35, 116, 45, 91, 87, 134, 63, 174, 71, 200, 125, 155, 87, 322, 125, 197, 172, 282, 113, 383, 131, 349, 217, 271, 213, 555, 161, 311, 267, 546, 183, 555, 195, 482, 402, 379, 215, 857, 267, 546, 369, 602, 245, 768, 349, 774, 421, 503, 281, 1204, 287, 561, 582, 875, 425

Offset: 1

Author

Ilya Gutkovskiy, Mar 21 2023

Keywords

• nonn

Comments

Dirichlet convolution of primes with the number of divisors function.

Links

• Winston de Greef, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000005, A000040, A007429, A007445, A361707.

Programs

Maple

a:= (proc(p) proc(n) uses numtheory;
       add(p(d), d=divisors(n))
     end end@@2)(ithprime):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 23 2023

Mathematica

Table[Sum[DivisorSigma[0, n/d] Prime[d], {d, Divisors[n]}], {n, 1, 65}]

PARI

a(n) = sumdiv(n, d, numdiv(n/d)*prime(d)); \\ Michel Marcus, Mar 23 2023

Formula

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

a(n) = Sum_{d|n} A000005(n/d) * prime(d).