A108223 - OEIS

%I #23 May 18 2024 14:52:58

%S 1,13,703,61681,9762501,2140365529,678222249307,280379743338241,

%T 150087010086914941,99902428887422922553,81402749386554449442711,

%U 79477293980103609858493681,91733330193268313783293023757,123469159731637675342948027295569,191751045863140709562160603031808243

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

%F 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.

%F From _Seiichi Manyama_, May 18 2024: (Start)

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

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

%e sigma_4(4)/sigma_2(4) =

%e (1 + 2^4 + 4^4)/(1 + 2^2 + 4^2) = 13.

%t Table[DivisorSigma[2n, n^2]/DivisorSigma[n, n^2], {n, 10}] (* _Ryan Propper_, Apr 03 2007 *)

%o (PARI) a(n) = sigma(n^2, 2*n)/sigma(n^2, n); \\ _Michel Marcus_, Sep 06 2019

%Y Cf. A062755.

%Y Cf. A057660, A084218, A084220, A372966.

%K nonn

%O 1,2

%A _Leroy Quet_, Jun 28 2005

%E More terms from _Ryan Propper_, Apr 03 2007

%E More terms from _Michel Marcus_, Sep 06 2019