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A351266
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Sum of the cubes of the squarefree divisors of n.
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11
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1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 9, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 9, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512, 103824
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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^3 * mu(d)^2.
G.f.: Sum_{k>=1} mu(k)^2 * k^3 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^3. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(4)/(4*zeta(2)) = Pi^2/60 = 0.164493... . - Amiram Eldar, Nov 10 2022
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EXAMPLE
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a(4) = 9; a(4) = Sum_{d|4} d^3 * mu(d)^2 = 1^3*1 + 2^3*1 + 4^3*0 = 9.
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^3); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
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PROG
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(PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^3)); \\ Michel Marcus, 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), this sequence (k=3), A351267 (k=4), A351268 (k=5), A351269 (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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