A344222 a(n) = Sum_{k=1..n} tau(gcd(k,n)^4), where tau(n) is the number of divisors of n.

1, 6, 7, 16, 9, 42, 11, 36, 25, 54, 15, 112, 17, 66, 63, 76, 21, 150, 23, 144, 77, 90, 27, 252, 49, 102, 79, 176, 33, 378, 35, 156, 105, 126, 99, 400, 41, 138, 119, 324, 45, 462, 47, 240, 225, 162, 51, 532, 81, 294, 147, 272, 57, 474, 135, 396, 161, 198, 63, 1008, 65, 210, 275, 316, 153, 630, 71

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^4).

a(n) = n * Sum_{d|n} 4^omega(d) / d.

If p is prime, a(p) = 4 + p.

From Amiram Eldar, Nov 15 2022: (Start)

Multiplicative with a(p^e) = (p^e*(p + 3) - 4)/(p - 1).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 3/p^2) = 2.4997873122... . (End)

Mathematica

Table[Sum[DivisorSigma[0, GCD[k, n]^4], {k, n}], {n, 100}] (* Giorgos Kalogeropoulos, May 13 2021 *)
f[p_, e_] := (p^e*(p + 3) - 4)/(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)^4));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*numdiv(d^4));
(PARI) a(n) = n*sumdiv(n, d, 4^omega(d)/d);

Crossrefs

Cf. A060648, A072691, A344221, A344223, A344224, A344225.

Sequence in context: A106680 A214325 A078388 * A315843 A261195 A008538

Adjacent sequences: A344219 A344220 A344221 * A344223 A344224 A344225

Keyword

nonn,mult

Author

Seiichi Manyama, May 12 2021

Status

approved