A371492 a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, n) )^2.

1, 17, 91, 289, 701, 1547, 2647, 4769, 7705, 11917, 15731, 26299, 30421, 44999, 63791, 77473, 87857, 130985, 136459, 202589, 240877, 267427, 290951, 433979, 448201, 517157, 633187, 764983, 729989, 1084447, 951391, 1248929, 1431521, 1493569, 1855547, 2226745

Offset

1,2

Links

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

Formula

a(n) = Sum_{d|n} phi(n/d) * (n/d)^2 * sigma_4(d^2)/sigma_2(d^2).

From Amiram Eldar, May 24 2024: (Start)

Multiplicative with a(p^e) = (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)).

Dirichlet g.f.: zeta(s)*zeta(s-3)*zeta(s-4)/zeta(s-2)^2.

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = zeta(2)*zeta(5)/zeta(3)^2 = 1.180448217... . (End)

Mathematica

f[p_, e_] := (p^(4*e+1)*(p+1)*(p^2+p+1) - p^(3*e+1)*(p^2+1) + p + 1)/((p^2+1)*(p^2+p+1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 36] (* Amiram Eldar, May 24 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^2*sigma(d^2, 4)/sigma(d^2, 2));

Crossrefs

Cf. A084218, A373060.

Cf. A372952, A372962.

Cf. A373059, A371628.

Cf. A002117, A013661, A013663.

Sequence in context: A159676 A061971 A061222 * A228462 A217641 A213574

Adjacent sequences: A371489 A371490 A371491 * A371493 A371494 A371495

Keyword

nonn,mult

Author

Seiichi Manyama, May 24 2024

Status

approved