A333557 a(n) = Sum_{d|n, gcd(d, n/d) = 1} uphi(d) * uphi(n/d), where uphi = unitary totient function (A047994).

1, 2, 4, 6, 8, 8, 12, 14, 16, 16, 20, 24, 24, 24, 32, 30, 32, 32, 36, 48, 48, 40, 44, 56, 48, 48, 52, 72, 56, 64, 60, 62, 80, 64, 96, 96, 72, 72, 96, 112, 80, 96, 84, 120, 128, 88, 92, 120, 96, 96, 128, 144, 104, 104, 160, 168, 144, 112, 116, 192, 120, 120, 192, 126, 192, 160, 132, 192, 176, 192

Offset: 1

Author

Ilya Gutkovskiy, Mar 26 2020

Keywords

• nonn
• mult

Links

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

Crossrefs

Cf. A001221, A029935, A034444, A047994, A055653, A062355, A145388.

Programs

Mathematica

uphi[1] = 1; uphi[n_] := Times @@ (#[[1]]^#[[2]] - 1 & /@ FactorInteger[n]); a[n_] := Sum[If[GCD[d, n/d] == 1, uphi[d] uphi[n/d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]
Table[Sum[If[GCD[d, n/d] == 1, (-2)^PrimeNu[n/d] 2^PrimeNu[d] d, 0], {d, Divisors[n]}], {n, 1, 70}]
f[p_, e_] := 2*(p^e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

PARI

a(n) = sumdiv(n, d, if (gcd(d, n/d) == 1, (-2)^omega(n/d)*2^omega(d)*d)); \\ Michel Marcus, Mar 27 2020

Formula

If n = Product (p_j^k_j) then a(n) = Product (2 * (p_j^k_j - 1)).

a(n) = 2^omega(n) * uphi(n).

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-2)^omega(n/d) * 2^omega(d) * d.

a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * A145388(d).