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A351269
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Sum of the 6th powers of the squarefree divisors of n.
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11
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1, 65, 730, 65, 15626, 47450, 117650, 65, 730, 1015690, 1771562, 47450, 4826810, 7647250, 11406980, 65, 24137570, 47450, 47045882, 1015690, 85884500, 115151530, 148035890, 47450, 15626, 313742650, 730, 7647250, 594823322, 741453700, 887503682, 65, 1293240260
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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^6 * mu(d)^2.
Multiplicative with a(p^e) = 1 + p^6. - Amiram Eldar, Feb 06 2022
G.f.: Sum_{k>=1} mu(k)^2 * k^6 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/(7*zeta(2)) = 0.0875718... . - Amiram Eldar, Nov 10 2022
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EXAMPLE
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a(4) = 65; a(4) = Sum_{d|4} d^6 * mu(d)^2 = 1^6*1 + 2^6*1 + 4^6*0 = 65.
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^6); 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), this sequence (k=6), A351270 (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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