cp's OEIS Frontend

A351305 a(n) = n^10 * Product_{p|n, p prime} (1 + 1/p^10).

%I A351305 #17 Dec 25 2024 02:01:13
%S A351305 1,1025,59050,1049600,9765626,60526250,282475250,1074790400,
%T A351305 3486843450,10009766650,25937424602,61978880000,137858491850,
%U A351305 289537131250,576660215300,1100585369600,2015993900450,3574014536250,6131066257802,10250001049600,16680163512500,26585860217050
%N A351305 a(n) = n^10 * Product_{p|n, p prime} (1 + 1/p^10).
%C A351305 Sum of the 10th powers of the divisor complements of the squarefree divisors of n.
%H A351305 Sebastian Karlsson, <a href="/A351305/b351305.txt">Table of n, a(n) for n = 1..10000</a>
%F A351305 a(n) = Sum_{d|n} d^10 * mu(n/d)^2.
%F A351305 a(n) = n^10 * Sum_{d|n} mu(d)^2 / d^10.
%F A351305 Multiplicative with a(p^e) = p^(10*e) + p^(10*e-10). - _Sebastian Karlsson_, Feb 08 2022
%F A351305 From _Vaclav Kotesovec_, Feb 12 2022: (Start)
%F A351305 Dirichlet g.f.: zeta(s)*zeta(s-10)/zeta(2*s).
%F A351305 Sum_{k=1..n} a(k) ~ n^11 * zeta(11) / (11 * zeta(22)) = 1222532449149375 * n^11 * zeta(11) / (155366 * Pi^22).
%F A351305 Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^10/(p^20-1)) = 1.000993621149252443797467720671490169127513829380371486971107300011796... (End)
%t A351305 f[p_, e_] := p^(10*e) + p^(10*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 20] (* _Amiram Eldar_, Feb 08 2022 *)
%o A351305 (PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^10);
%o A351305 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^10*X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 12 2022
%Y A351305 Cf. A008683 (mu).
%Y A351305 Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), A065959 (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), this sequence (k=10).
%K A351305 nonn,mult
%O A351305 1,2
%A A351305 _Wesley Ivan Hurt_, Feb 06 2022