A341636 a(n) = Sum_{d|n} phi(d) * tau(d) * tau(n/d).

1, 4, 6, 13, 10, 24, 14, 38, 29, 40, 22, 78, 26, 56, 60, 103, 34, 116, 38, 130, 84, 88, 46, 228, 79, 104, 124, 182, 58, 240, 62, 264, 132, 136, 140, 377, 74, 152, 156, 380, 82, 336, 86, 286, 290, 184, 94, 618, 153, 316, 204, 338, 106, 496, 220, 532, 228, 232, 118, 780, 122, 248

Offset: 1

Author

Ilya Gutkovskiy, Feb 16 2021

Keywords

• nonn
• easy
• mult

Comments

Inverse Moebius transform of A062949.

Links

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

Crossrefs

Cf. A000005, A000010, A000203, A007426, A055507, A062355, A062949.

Programs

Mathematica

Table[Sum[EulerPhi[d] DivisorSigma[0, d] DivisorSigma[0, n/d], {d, Divisors[n]}], {n, 62}]
Table[Sum[DivisorSigma[0, GCD[n, k]] DivisorSigma[0, n/GCD[n, k]], {k, n}], {n, 62}]
f[p_, e_] := (p + 1 + e*(p - 1) + p^(e + 1)*(e*(p - 1) + p - 3))/(p - 1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2023 *)

PARI

a(n) = sumdiv(n, d, eulerphi(d)*numdiv(d)*numdiv(n/d)); \\ Michel Marcus, Feb 17 2021

Formula

a(n) = Sum_{k=1..n} tau(gcd(n,k)) * tau(n/gcd(n,k)).

a(n) = Sum_{d|n} A062949(d).

Multiplicative with a(p^e) = (p + 1 + e*(p-1) + p^(e+1)*(e*(p-1)+p-3))/(p-1)^2. - Amiram Eldar, Sep 15 2023