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

Author

Vladeta Jovovic, Jul 07 2001

Keywords

• nonn
• mult

Comments

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

Examples

			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
		

Links

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

Crossrefs

Cf. A000005, A000010, A060648, A062949.

Programs

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 *)

PARI

a(n)=sumdiv(n,i,eulerphi(i)*sumdiv(n,j,eulerphi(j)/eulerphi(gcd(i,j)))) \\ Charles R Greathouse IV, Sep 12 2012

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

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