%I #39 May 19 2024 09:39:02

%S 1,4,7,14,13,28,19,42,37,52,31,98,37,76,91,114,49,148,55,182,133,124,

%T 67,294,113,148,163,266,85,364,91,290,217,196,247,518,109,220,259,546,

%U 121,532,127,434,481,268,139,798,229,452,343,518,157,652,403,798,385

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

%C A176003 is a subsequence. - _Peter Luschny_, Sep 12 2012

%H Charles R Greathouse IV, <a href="/A062380/b062380.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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

%F a(n) = Sum_{c|n,d|n} phi(lcm(c,d)). - _Peter Luschny_, Sep 10 2012

%F a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^2 ). - _Seiichi Manyama_, May 19 2024

%e 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

%p with(numtheory):

%p a:= n-> add(phi(d)*tau(d^2), d=divisors(n)):

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Sep 12 2012

%t a[n_] := DivisorSum[n, EulerPhi[#] DivisorSigma[0, #^2]&]; Array[a, 60] (* _Jean-François Alcover_, Dec 05 2015 *)

%t 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 *)

%o (Sage)

%o def A062380(n) :

%o d = divisors(n); cp = cartesian_product([d, d])

%o return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))

%o [A062380(n) for n in (1..57)] # _Peter Luschny_, Sep 10 2012

%o (PARI) a(n)=sumdiv(n,i,eulerphi(i)*sumdiv(n,j,eulerphi(j)/eulerphi(gcd(i,j)))) \\ _Charles R Greathouse IV_, Sep 12 2012

%Y Cf. A000005, A000010, A060648, A062949.

%K nonn,mult

%O 1,2

%A _Vladeta Jovovic_, Jul 07 2001