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A351270
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Sum of the 7th powers of the squarefree divisors of n.
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11
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1, 129, 2188, 129, 78126, 282252, 823544, 129, 2188, 10078254, 19487172, 282252, 62748518, 106237176, 170939688, 129, 410338674, 282252, 893871740, 10078254, 1801914272, 2513845188, 3404825448, 282252, 78126, 8094558822, 2188, 106237176, 17249876310, 22051219752, 27512614112
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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^7 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^7. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^7 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/(8*zeta(2)) = Pi^6/12600 = 0.0763007... . - Amiram Eldar, Nov 10 2022
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EXAMPLE
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a(4) = 129; a(4) = Sum_{d|4} d^7 * mu(d)^2 = 1^7*1 + 2^7*1 + 4^7*0 = 129.
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^7); 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), this sequence (k=7), A351271 (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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