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A351267
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Sum of the 4th powers of the squarefree divisors of n.
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11
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1, 17, 82, 17, 626, 1394, 2402, 17, 82, 10642, 14642, 1394, 28562, 40834, 51332, 17, 83522, 1394, 130322, 10642, 196964, 248914, 279842, 1394, 626, 485554, 82, 40834, 707282, 872644, 923522, 17, 1200644, 1419874, 1503652, 1394, 1874162, 2215474, 2342084, 10642, 2825762, 3348388
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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^4 * mu(d)^2.
G.f.: Sum_{k>=1} mu(k)^2 * k^4 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022
Multiplicative with a(p^e) = 1 + p^4. - Amiram Eldar, Feb 06 2022
Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Nov 10 2022
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EXAMPLE
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a(4) = 17; a(4) = Sum_{d|4} d^4 * mu(d)^2 = 1^4*1 + 2^4*1 + 4^4*0 = 17.
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MATHEMATICA
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a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^4); 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^4)); \\ 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), A351266 (k=3), this sequence (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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