A371878 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).

1, 63, 727, 4031, 15621, 45801, 117643, 257983, 529981, 984123, 1771551, 2930537, 4826797, 7411509, 11356467, 16510911, 24137553, 33388803, 47045863, 62968251, 85526461, 111607713, 148035867, 187553641, 244078121, 304088211, 386356147, 474218933, 594823293, 715457421

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, x_2, x_3, x_4, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.

a(n) = Sum_{d|n} mu(n/d) * (n/d) * sigma_6(d).

From Amiram Eldar, May 25 2024: (Start)

Multiplicative with a(p^e) = (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1).

Dirichlet g.f.: zeta(s)*zeta(s-6)/zeta(s-1).

Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = zeta(7)/zeta(6) = 0.9911595361106... . (End)

Mathematica

f[p_, e_] := (p^(6*e+6) - p^(6*e+1) + p - 1)/(p^6-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, May 25 2024 *)

Prog

(PARI) a(n) = sumdiv(n, d, moebius(n/d)*n/d*sigma(d, 6));

Crossrefs

Column k=5 of A372968.

Cf. A371491, A373007, A373103, A373105.

Cf. A013664, A013665.

Sequence in context: A221968 A115152 A343978 * A284953 A069091 A123866

Adjacent sequences: A371875 A371876 A371877 * A371879 A371880 A371881

Keyword

nonn,mult

Author

Seiichi Manyama, May 25 2024

Status

approved