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A078429
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Number of integers k among 1..n for which gcd(k,n) is a cube.
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3
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1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 9, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 18, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 18, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 37, 48, 20, 66, 32, 44, 24, 70, 30, 72, 36, 40, 36
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OFFSET
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1,3
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LINKS
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FORMULA
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a(n) is multiplicative.
G.f. for a(p^n), p a prime, is given by 1/(1+x+x^2)/(1-p*x).
a(p) = p-1, a(p^2) = p*(p-1), a(p^3) = p^3-p^2+1, a(p^4) = (p-1)*(p+1)*(p^2-p+1), ...
a(n) = Sum_{d|n, d is a perfect cube} phi(n/d), where phi(k) is the Euler totient function. Dirichlet convolution of A000010 and A010057. - Daniel Suteu, Jun 27 2018
a(n) = Sum_{k=1..n} A010057(gcd(n,k)).
a(n) = Sum_{k=1..n} A010057(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
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MATHEMATICA
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nn = 76; f[list_, i_] := list[[i]]; a = Table[If[IntegerQ[n^(1/3)], 1, 0], {n, 1, nn}]; b =Table[EulerPhi[n], {n, 1, nn}]; Table[DirichletConvolve[f[a, n], f[b, n], n, m], {m, 1, nn}] (* Geoffrey Critzer, Feb 25 2015 *)
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PROG
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(PARI) a(n) = sum(k=1, n, ispower(gcd(n, k), 3)); \\ Michel Marcus, Feb 25 2015
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d) * ispower(d, 3)); \\ Daniel Suteu, Jun 27 2018
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CROSSREFS
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KEYWORD
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mult,nonn
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AUTHOR
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STATUS
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approved
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