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URL: https://oeis.org/A062380

⇱ A062380 - OEIS

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A062380
a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.
13
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
(SageMath)
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
KEYWORD
nonn,mult
AUTHOR
Vladeta Jovovic, Jul 07 2001
STATUS
approved