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A351271
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Sum of the 8th powers of the squarefree divisors of n.
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11
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1, 257, 6562, 257, 390626, 1686434, 5764802, 257, 6562, 100390882, 214358882, 1686434, 815730722, 1481554114, 2563287812, 257, 6975757442, 1686434, 16983563042, 100390882, 37828630724, 55090232674, 78310985282, 1686434, 390626, 209642795554, 6562, 1481554114, 500246412962
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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a(n) = Sum_{d|n} d^8 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^8. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^8 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(9*zeta(2)) = 0.0676831... . - Amiram Eldar, Nov 10 2022
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EXAMPLE
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a(4) = 257; a(4) = Sum_{d|4} d^8 * mu(d)^2 = 1^8*1 + 2^8*1 + 4^8*0 = 257.
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^8); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
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CROSSREFS
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Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), this sequence (k=8), A351272 (k=9), A351273 (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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