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A254981
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a(n) is the sum of the divisors d of n such that n/d is cubefree.
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5
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1, 3, 4, 7, 6, 12, 8, 14, 13, 18, 12, 28, 14, 24, 24, 28, 18, 39, 20, 42, 32, 36, 24, 56, 31, 42, 39, 56, 30, 72, 32, 56, 48, 54, 48, 91, 38, 60, 56, 84, 42, 96, 44, 84, 78, 72, 48, 112, 57, 93, 72, 98, 54, 117, 72, 112, 80, 90, 60, 168, 62, 96, 104, 112, 84, 144
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OFFSET
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1,2
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COMMENTS
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Inverse Möbius transform of A254926.
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LINKS
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Table of n, a(n) for n = 1..10000
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FORMULA
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a(n) = Sum_{d | n} d * A212793(n/d) = n * Sum_{d | n} A212793(d) / d.
a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3).
Multiplicative with a(p) = 1 + p; a(p^e) = p^(e-2) * (1 + p + p^2), for e>1.
Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(3s).
If n is powerful, a(n^k) = n^(k-1) * a(n).
For k>1, a(n^k) = n^(k-1) * a(n) * Prod_{p prime, ord(n,p)=1} (p^3-1) / (p^3-p).
Sum_{k=1..n} a(k) ~ 315*n^2 / (4*Pi^4). - Vaclav Kotesovec, Feb 03 2019
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MATHEMATICA
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nn = 66; f[list_, i_] := list[[i]]; a = Table[If[Max[FactorInteger[n][[All, 2]]] < 3, 1, 0], {n, 1, nn}]; b =Table[n, {n, 1, nn}]; Table[
DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 22 2015 *)
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PROG
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(PARI) a212793(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 2]) >=3, return(0)); ); return (1); }
a(n) = sumdiv(n, d, d*a212793(n/d)); \\ Michel Marcus, Feb 11 2015
(PARI) a(n) = sumdiv(n, d, if (ispower(d, 3), moebius(sqrtnint(d, 3))*sigma(n/d), 0)); \\ Michel Marcus, Mar 04 2015
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CROSSREFS
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Cf. A000203, A001615, A001694, A212793, A254926.
Sequence in context: A333926 A051378 A344575 * A116607 A107749 A093811
Adjacent sequences: A254978 A254979 A254980 * A254982 A254983 A254984
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KEYWORD
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mult,nonn
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AUTHOR
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Álvar Ibeas, Feb 11 2015
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STATUS
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approved
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