A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

1, 3, 3, 5, 3, 9, 3, 8, 5, 9, 3, 15, 3, 9, 9, 11, 3, 15, 3, 15, 9, 9, 3, 24, 5, 9, 8, 15, 3, 27, 3, 15, 9, 9, 9, 25, 3, 9, 9, 24, 3, 27, 3, 15, 15, 9, 3, 33, 5, 15, 9, 15, 3, 24, 9, 24, 9, 9, 3, 45, 3, 9, 15, 19, 9, 27, 3, 15, 9, 27, 3, 40, 3, 9, 15

Offset: 1

Author

Ilya Gutkovskiy, Feb 21 2020

Keywords

• nonn
• mult

Comments

Inverse Moebius transform of A322483.

Links

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

Crossrefs

Cf. A000005, A007425, A024206, A048691, A124315, A124316, A295316, A322483, A332619, A332713.

Programs

Mathematica

Table[Sum[DivisorSigma[0, d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e+3)/2]; A322483[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Table[Sum[A322483[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e + 1)*(e + 5)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

Formula

a(n) = Sum_{d|n} A322483(d).

a(n) = Sum_{d|n} tau(n/d) * A295316(d).

Multiplicative with a(p^e) = floor((e+1)*(e+5)/4) = A024206(e+2). - Amiram Eldar, Dec 05 2022