A069208 a(n) = Sum_{ d divides n } phi(n)/phi(d).

1, 2, 3, 5, 5, 6, 7, 11, 10, 10, 11, 15, 13, 14, 15, 23, 17, 20, 19, 25, 21, 22, 23, 33, 26, 26, 31, 35, 29, 30, 31, 47, 33, 34, 35, 50, 37, 38, 39, 55, 41, 42, 43, 55, 50, 46, 47, 69, 50, 52, 51, 65, 53, 62, 55, 77, 57, 58, 59, 75, 61, 62, 70, 95, 65, 66, 67, 85, 69, 70, 71

Offset: 1

Author

Vladeta Jovovic, Apr 10 2002

Keywords

• nonn
• mult

Comments

a(n) = n iff n is squarefree number (cf. A005117).

Conjecture: Let (f(n)), n > 0, be a multiplicative sequence. Then holds:

(1) p(f; n) = Sum_{d powerful number (A001694) dividing n} f(d) is multiplicative;

(2) p(f; n) equals inverse Moebius transform of A112526(n) * f(n). - Werner Schulte, Jan 23 2025

a(n) is also the number of conjugacy classes of the holomorph of the cyclic group of order n. Corollary: Let Rn be the dihedral quandle of order n. Then a(n) is the number of isomorphism classes of virtual quandles whose underlying quandle is isomorphic to Rn. - Luc Ta, Jun 16 2025

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 2.

Crossrefs

Cf. A000010, A000027, A005117, A005361, A069170, A112526.

Programs

Mathematica

Table[EulerPhi[n]*Total[1/EulerPhi@Divisors@n], {n, 71}] (* Ivan Neretin, Sep 20 2017 *)
f[p_, e_] := (p^(e + 1) - p^e + p^(e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 14 2022 *)

PARI

a(n) = sumdiv(n, d, eulerphi(n)/eulerphi(d)) \\ Michel Marcus, Jun 17 2013

PARI

a(n) = my(f=factor(n)); prod(k=1, #f~, (f[k,1]^(f[k,2]-1) + (f[k,1]-1)*f[k,1]^f[k,2]-1) / (f[k,1]-1)); \\ Daniel Suteu, Nov 04 2018

Formula

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

a(n) = phi(n) * Sum_{k=1..n} 1/phi(n / gcd(n, k))^2. - Daniel Suteu, Nov 04 2018

a(n) = Sum_{k=1..n, gcd(n,k) = 1} tau(gcd(n,k-1)). - Ilya Gutkovskiy, Sep 24 2021

From Werner Schulte, Feb 27 2022: (Start)

Dirichlet convolution of A005361 and A000010.

Dirichlet convolution of A112526 and A000027.

Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-1) * zeta(2*s) * zeta(3*s) / zeta(6*s). (End)

Sum_{k=1..n} a(k) ~ c * n^2, where c = 15015/(2764*Pi^2) = 0.550411... . - Amiram Eldar, Oct 22 2022

a(n) = Sum_{d powerful number (A001694) dividing n} n / d. - Werner Schulte, Jan 23 2025 (see Golomb link at A001694)