A344221 a(n) = Sum_{k=1..n} tau(gcd(k,n)^3), where tau(n) is the number of divisors of n.

1, 5, 6, 13, 8, 30, 10, 29, 21, 40, 14, 78, 16, 50, 48, 61, 20, 105, 22, 104, 60, 70, 26, 174, 43, 80, 66, 130, 32, 240, 34, 125, 84, 100, 80, 273, 40, 110, 96, 232, 44, 300, 46, 182, 168, 130, 50, 366, 73, 215, 120, 208, 56, 330, 112, 290, 132, 160, 62, 624, 64, 170, 210, 253, 128, 420

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} phi(n/d) * tau(d^3).

a(n) = n * Sum_{d|n} 3^omega(d) / d.

If p is prime, a(p) = 3 + p.

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = (p^e*(p + 2) - 3)/(p - 1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 2/p^2) = 1.8019184198... . (End)

Mathematica

Table[Sum[DivisorSigma[0, GCD[k, n]^3], {k, n}], {n, 100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 2) - 3)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 15 2022 *)

Prog

(PARI) a(n) = sum(k=1, n, numdiv(gcd(k, n)^3));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^3));
(PARI) a(n) = n*sumdiv(n, d, 3^omega(d)/d);

Crossrefs

Cf. A060648, A072691, A344222, A344223, A344224, A344225.

Sequence in context: A173074 A099473 A051572 * A348155 A047436 A334320

Adjacent sequences: A344218 A344219 A344220 * A344222 A344223 A344224

Keyword

nonn,mult

Author

Seiichi Manyama, May 12 2021

Status

approved