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A351300
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a(n) = n^5 * Product_{p|n, p prime} (1 + 1/p^5).
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11
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1, 33, 244, 1056, 3126, 8052, 16808, 33792, 59292, 103158, 161052, 257664, 371294, 554664, 762744, 1081344, 1419858, 1956636, 2476100, 3301056, 4101152, 5314716, 6436344, 8245248, 9768750, 12252702, 14407956, 17749248, 20511150, 25170552, 28629152, 34603008, 39296688
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OFFSET
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1,2
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COMMENTS
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Sum of the 5th powers of the divisor complements of the squarefree divisors of n.
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LINKS
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FORMULA
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a(n) = Sum_{d|n} d^5 * mu(n/d)^2.
a(n) = n^5 * Sum_{d|n} mu(d)^2 / d^5.
Dirichlet g.f.: zeta(s)*zeta(s-5)/zeta(2*s).
Sum_{k=1..n} a(k) ~ n^6 * zeta(6) / (6 * zeta(12)) = 225225 * n^6 / (1382 * Pi^6).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^5/(p^10-1)) = 1.03592823428850098309076014982275428113698561633329794485946580153004... (End)
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MATHEMATICA
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f[p_, e_] := p^(5*e) + p^(5*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Feb 08 2022 *)
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PROG
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(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^5);
(PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + X)/(1 - p^5*X))[n], ", ")) \\ Vaclav Kotesovec, Feb 12 2022
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CROSSREFS
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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), this sequence (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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