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

1,3

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).

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(2^n) = A077947(n), a(3^n) = A077834(n).

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

Dirichlet convolution of A000027 and A210826. - R. J. Mathar, Jun 05 2020

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)

Multiplicative with a(p^e) = Sum_{k=0..floor(e/3)} phi(p^(e-3*k)). - Amiram Eldar, Nov 15 2025

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 *)
f[p_, e_] := Sum[If[3*k == e, 1, (p-1)*p^(e-3*k-1)], {k, 0, Floor[e/3]}];
a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Nov 15 2025 *)

Prog

(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
(PARI) s(p, e) = sum(k = 0, e\3, if(3*k == e, 1, (p-1)*p^(e-3*k-1)));
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i, 1], f[i, 2])); } \\ Amiram Eldar, Nov 15 2025

Crossrefs

Cf. A061020, A206369, A327626 (inv. Mob. Trans.).

Cf. A000010, A000027, A010057, A077834, A077947, A210826.

Sequence in context: A054929 A236628 A390606 * A384040 A360523 A358039

Adjacent sequences: A078426 A078427 A078428 * A078430 A078431 A078432

Keyword

mult,nonn,easy

Author

Vladeta Jovovic, Dec 29 2002

Status

approved