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a(n) = n^7 * Product_{p|n, p prime} (1 + 1/p^7).
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%I #21 Nov 14 2022 05:35:08

%S 1,129,2188,16512,78126,282252,823544,2113536,4785156,10078254,

%T 19487172,36128256,62748518,106237176,170939688,270532608,410338674,

%U 617285124,893871740,1290016512,1801914272,2513845188,3404825448,4624416768,6103593750,8094558822,10465136172

%N a(n) = n^7 * Product_{p|n, p prime} (1 + 1/p^7).

%C Sum of the 7th powers of the divisor complements of the squarefree divisors of n.

%H Sebastian Karlsson, <a href="/A351302/b351302.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} d^7 * mu(n/d)^2.

%F a(n) = n^7 * Sum_{d|n} mu(d)^2 / d^7.

%F Multiplicative with a(p^e) = p^(7*e) + p^(7*e-7). - _Sebastian Karlsson_, Feb 08 2022

%F From _Vaclav Kotesovec_, Feb 12 2022: (Start)

%F Dirichlet g.f.: zeta(s)*zeta(s-7)/zeta(2*s).

%F Sum_{k=1..n} a(k) ~ n^8 * zeta(8) / (8 * zeta(16)) = 34459425 * n^8 / (28936 * Pi^8).

%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^7/(p^14-1)) = 1.008287998838997802253937842472728682107868602338715231926150271159410... (End)

%F a(n) = J_14(n) / J_7(n) = J_14(n) / A069092(n), where J_k is the k-th Jordan totient function. - _Enrique PĂ©rez Herrero_, Nov 13 2022

%t f[p_, e_] := p^(7*e) + p^(7*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* _Amiram Eldar_, Feb 08 2022 *)

%o (PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^7);

%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^7*X))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 12 2022

%Y Cf. A008683 (mu), A069092.

%Y 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), this sequence (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 06 2022