A373103 a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( n/gcd(x_1, x_2, x_3, x_4, x_5, n) )^4.

1, 497, 19603, 254449, 1952501, 9742691, 40351207, 130277873, 385845769, 970392997, 2357933051, 4987963747, 10604470813, 20054549879, 38274877103, 66702270961, 118587792977, 191765347193, 322687567459, 496811926949, 791004710821, 1171892726347, 1801152381623

Offset

1,2

Links

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

Formula

a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.

a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_9(d).

a(n) = Sum_{d|n} phi(n/d) * (n/d)^8 * sigma_8(d^2)/sigma_4(d^2).

From Amiram Eldar, May 25 2024: (Start)

Multiplicative with a(p^e) = (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1).

Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(s-4).

Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = zeta(10)/zeta(6) = Pi^4/99 = 0.983930212464... . (End)

Mathematica

f[p_, e_] :=  (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 25 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 9));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^8*sigma(d^2, 8)/sigma(d^2, 4));

Crossrefs

Cf. A371491, A371878, A373007, A373105.

Cf. A350156, A372962, A372964.

Cf. A013664, A013668.

Sequence in context: A097785 A145922 A145981 * A210785 A064255 A067917

Adjacent sequences: A373096 A373101 A373102 * A373104 A373105 A373106

Keyword

nonn,mult

Author

Seiichi Manyama, May 25 2024

Status

approved