A078429 Number of integers k among 1..n for which gcd(k,n) is a cube.
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
Offset: 1
Links
- Daniel Suteu, Table of n, a(n) for n = 1..10000
- Eckford Cohen, A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where a(n) = Psi_3(n).
Crossrefs
Programs
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Mathematica
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 *) -
PARI
a(n) = sum(k=1, n, ispower(gcd(n, k), 3)); \\ Michel Marcus, Feb 25 2015
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PARI
a(n) = sumdiv(n, d, eulerphi(n/d) * ispower(d, 3)); \\ Daniel Suteu, Jun 27 2018
Formula
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), ...
Dirichlet g.f.: zeta(s - 1)*zeta(3*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
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
Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 315. - Vaclav Kotesovec, Feb 07 2019
From Richard L. Ollerton, May 07 2021: (Start)
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)