A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.

1, 4, 7, 14, 13, 28, 19, 42, 37, 52, 31, 98, 37, 76, 91, 114, 49, 148, 55, 182, 133, 124, 67, 294, 113, 148, 163, 266, 85, 364, 91, 290, 217, 196, 247, 518, 109, 220, 259, 546, 121, 532, 127, 434, 481, 268, 139, 798, 229, 452, 343, 518, 157, 652, 403, 798, 385

Offset

1,2

Comments

A176003 is a subsequence. - Peter Luschny, Sep 12 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d|n} phi(d)*tau(d^2).

Multiplicative with a(p^e) = 1 + Sum_{k=1..e} (2k+1)(p^k-p^{k-1}) = ((2e+1)p^(e+1) - (2e+3)p^e+2)/(p-1). - Mitch Harris, May 24 2005

a(n) = Sum_{c|n,d|n} phi(lcm(c,d)). - Peter Luschny, Sep 10 2012

a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^2 ). - Seiichi Manyama, May 19 2024

Example

Let p be a prime then a(p) = phi(1)*tau(1)+phi(p)*tau(p^2) = 1+(p-1)*3 = 3*p-2. - Peter Luschny, Sep 12 2012

Maple

with(numtheory):
a:= n->  add(phi(d)*tau(d^2), d=divisors(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Sep 12 2012

Mathematica

a[n_] := DivisorSum[n, EulerPhi[#] DivisorSigma[0, #^2]&]; Array[a, 60] (* Jean-François Alcover, Dec 05 2015 *)
f[p_, e_] := ((2*e+1)*p^(e+1) - (2*e+3)*p^e + 2)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

Prog

(Sage)
def A062380(n) :
    d = divisors(n); cp = cartesian_product([d, d])
    return reduce(lambda x, y: x+y, map(euler_phi, map(lcm, cp)))
[A062380(n) for n in (1..57)]  # Peter Luschny, Sep 10 2012
(PARI) a(n)=sumdiv(n, i, eulerphi(i)*sumdiv(n, j, eulerphi(j)/eulerphi(gcd(i, j)))) \\ Charles R Greathouse IV, Sep 12 2012

Crossrefs

Cf. A000005, A000010, A060648, A062949.

Sequence in context: A347137 A347413 A310825 * A310826 A310827 A310828

Adjacent sequences: A062377 A062378 A062379 * A062381 A062382 A062383

Keyword

nonn,mult

Author

Vladeta Jovovic, Jul 07 2001

Status

approved