A068963 a(n) = Sum_{d|n} phi(d^3).

1, 5, 19, 37, 101, 95, 295, 293, 505, 505, 1211, 703, 2029, 1475, 1919, 2341, 4625, 2525, 6499, 3737, 5605, 6055, 11639, 5567, 12601, 10145, 13627, 10915, 23549, 9595, 28831, 18725, 23009, 23125, 29795, 18685, 49285, 32495, 38551, 29593

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Formula

Also Sum_{d|n} d*phi(d^2), or Sum_{d|n} d^2*phi(d).

Also Sum_{k=1..n} (n/gcd(n, k))^2 = Sum_{k=1..n} (lcm(n, k)/k)^2. - Vladeta Jovovic, Dec 29 2002

Multiplicative with a(p^e) = 1 + p^2 * (p-1)*(p^(3e)-1)/(p^3-1).

G.f.: Sum_{k>=1} k^2*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018

Dirichlet g.f.: Sum_{n>=1} a(n) / n^s = zeta(s) * zeta(s-3) / zeta(s-2). - Werner Schulte, Feb 18 2021

Sum_{k=1..n} a(k) ~ Pi^2 * n^4 / 60. - Vaclav Kotesovec, Aug 20 2021

Mathematica

Table[Total[EulerPhi[Divisors[n]^3]], {n, 50}] (* Harvey P. Dale, Feb 24 2013 *)
f[p_, e_] := p^2*(p - 1)*(p^(3 e) - 1)/(p^3 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Jun 19 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d^3)); \\ Michel Marcus, Mar 10 2018

Crossrefs

Cf. A000010, A057660, A056789.

Sequence in context: A297750 A285226 A146861 * A257929 A254060 A129828

Adjacent sequences: A068960 A068961 A068962 * A068964 A068965 A068966

Keyword

easy,nonn,mult

Author

Benoit Cloitre, Apr 06 2002

Status

approved