This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A078429 #34 May 07 2021 12:34:17
%S A078429 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,
%T A078429 28,8,30,18,20,16,24,12,36,18,24,20,40,12,42,20,24,22,46,18,42,20,32,
%U A078429 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
%N A078429 Number of integers k among 1..n for which gcd(k,n) is a cube.
%H A078429 Daniel Suteu, <a href="/A078429/b078429.txt">Table of n, a(n) for n = 1..10000</a>
%H A078429 Eckford Cohen, <a href="https://msp.org/pjm/1959/9-1/pjm-v9-n1-s.pdf">A class of residue systems (mod r) and related arithmetical functions. I. A generalization of the Moebius function</a>, Pacific J. Math. 9(1) (1959), 13-24; see Section 6 where a(n) = Psi_3(n).
%F A078429 a(n) is multiplicative.
%F A078429 G.f. for a(p^n), p a prime, is given by 1/(1+x+x^2)/(1-p*x).
%F A078429 a(2^n) = A077947(n), a(3^n) = A077834(n).
%F A078429 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), ...
%F A078429 Dirichlet g.f.: zeta(s - 1)*zeta(3*s)/zeta(s). - _Geoffrey Critzer_, Feb 25 2015
%F A078429 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
%F A078429 Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 315. - _Vaclav Kotesovec_, Feb 07 2019
%F A078429 Dirichlet convolution of A000027 and A210826. - _R. J. Mathar_, Jun 05 2020
%F A078429 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A078429 a(n) = Sum_{k=1..n} A010057(gcd(n,k)).
%F A078429 a(n) = Sum_{k=1..n} A010057(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%t A078429 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 *)
%o A078429 (PARI) a(n) = sum(k=1, n, ispower(gcd(n, k), 3)); \\ _Michel Marcus_, Feb 25 2015
%o A078429 (PARI) a(n) = sumdiv(n, d, eulerphi(n/d) * ispower(d, 3)); \\ _Daniel Suteu_, Jun 27 2018
%Y A078429 Cf. A061020, A206369, A327626 (inv. Mob. Trans.).
%Y A078429 Cf. A000010, A000027, A010057, A077834, A077947, A210826.
%K A078429 mult,nonn
%O A078429 1,3
%A A078429 _Vladeta Jovovic_, Dec 29 2002