A068970 a(n) = Sum_{d|n} phi(d^4).

1, 9, 55, 137, 501, 495, 2059, 2185, 4429, 4509, 13311, 7535, 26365, 18531, 27555, 34953, 78609, 39861, 123463, 68637, 113245, 119799, 267675, 120175, 313001, 237285, 358723, 282083, 682893, 247995, 893731, 559241, 732105, 707481, 1031559, 606773, 1823509

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

Also Sum_{d|n} d^m*phi(d^(4-m)) for m=0, 1, 2, 3.

Multiplicative with a(p^e) = 1 + p^3 * (p-1)(p^(4e)-1)/(p^4-1).

G.f.: Sum_{k>=1} k^3*phi(k)*x^k/(1 - x^k). - Ilya Gutkovskiy, Mar 10 2018

Sum_{k=1..n} a(k) ~ c * n^5, where c = zeta(5)/(5*zeta(2)) = 0.126075... . - Amiram Eldar, Dec 01 2022

From Peter Bala, Jan 21 2024: (Start)

a(n) = Sum_{k = 1..n} (n/gcd(k, n))^3 = Sum_{k = 1..n} (lcm(k, n)/k)^3.

Dirichlet g.f.: zeta(s) * zeta(s-4)/zeta(s-3). (End)

Maple

f:= n -> add(numtheory:-phi(d^4), d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Sep 13 2018

Mathematica

Table[Total[EulerPhi[Divisors[n]^4]], {n, 40}] (* Vincenzo Librandi, Sep 13 2018 *)
f[p_, e_] := 1 + p^3*(p - 1)*(p^(4*e) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Dec 01 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, eulerphi(d^4)); \\ Michel Marcus, Mar 10 2018

Crossrefs

Cf. A000010, A013661, A013663, A057660, A068963, A189393.

Sequence in context: A058852 A145875 A299519 * A141530 A263478 A326249

Adjacent sequences: A068967 A068968 A068969 * A068971 A068972 A068973

Keyword

easy,nonn,mult

Author

Benoit Cloitre, Apr 06 2002

Status

approved