A108223 a(n) = sigma_{2n}(n^2)/sigma_n(n^2), where sigma_n(m) = Sum_{d|m} d^n.

1, 13, 703, 61681, 9762501, 2140365529, 678222249307, 280379743338241, 150087010086914941, 99902428887422922553, 81402749386554449442711, 79477293980103609858493681, 91733330193268313783293023757, 123469159731637675342948027295569, 191751045863140709562160603031808243

Offset

1,2

Links

Table of n, a(n) for n=1..15.

Formula

a(n) = Product_{p=primes} (Sum_{k=0..2*b(n, p)} p^(n*k)*(-1)^k), where p^b(n, p) is the highest power of p dividing n.

From Seiichi Manyama, May 18 2024: (Start)

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

a(n) = Sum_{d|n} mu(n/d) * (n/d)^n * sigma_{2*n}(d). (End)

Example

sigma_4(4)/sigma_2(4) =
(1 + 2^4 + 4^4)/(1 + 2^2 + 4^2) = 13.

Mathematica

Table[DivisorSigma[2n, n^2]/DivisorSigma[n, n^2], {n, 10}] (* Ryan Propper, Apr 03 2007 *)

Prog

(PARI) a(n) = sigma(n^2, 2*n)/sigma(n^2, n); \\ Michel Marcus, Sep 06 2019

Crossrefs

Cf. A062755.

Cf. A057660, A084218, A084220, A372966.

Sequence in context: A113093 A195576 A195555 * A114355 A297748 A086157

Adjacent sequences: A108220 A108221 A108222 * A108224 A108225 A108226

Keyword

nonn,changed

Author

Leroy Quet, Jun 28 2005

Extensions

More terms from Ryan Propper, Apr 03 2007

More terms from Michel Marcus, Sep 06 2019

Status

approved