This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A062380 #39 May 19 2024 09:39:02
%S A062380 1,4,7,14,13,28,19,42,37,52,31,98,37,76,91,114,49,148,55,182,133,124,
%T A062380 67,294,113,148,163,266,85,364,91,290,217,196,247,518,109,220,259,546,
%U A062380 121,532,127,434,481,268,139,798,229,452,343,518,157,652,403,798,385
%N A062380 a(n) = Sum_{i|n,j|n} phi(i)*phi(j)/phi(gcd(i,j)), where phi is Euler totient function.
%C A062380 A176003 is a subsequence. - _Peter Luschny_, Sep 12 2012
%H A062380 Charles R Greathouse IV, <a href="/A062380/b062380.txt">Table of n, a(n) for n = 1..10000</a>
%F A062380 a(n) = Sum_{d|n} phi(d)*tau(d^2).
%F A062380 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 A062380 a(n) = Sum_{c|n,d|n} phi(lcm(c,d)). - _Peter Luschny_, Sep 10 2012
%F A062380 a(n) = Sum_{k=1..n} tau( (n/gcd(k,n))^2 ). - _Seiichi Manyama_, May 19 2024
%e A062380 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 A062380 with(numtheory):
%p A062380 a:= n-> add(phi(d)*tau(d^2), d=divisors(n)):
%p A062380 seq(a(n), n=1..60); # _Alois P. Heinz_, Sep 12 2012
%t A062380 a[n_] := DivisorSum[n, EulerPhi[#] DivisorSigma[0, #^2]&]; Array[a, 60] (* _Jean-François Alcover_, Dec 05 2015 *)
%t A062380 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 A062380 (Sage)
%o A062380 def A062380(n) :
%o A062380 d = divisors(n); cp = cartesian_product([d, d])
%o A062380 return reduce(lambda x,y: x+y, map(euler_phi, map(lcm, cp)))
%o A062380 [A062380(n) for n in (1..57)] # _Peter Luschny_, Sep 10 2012
%o A062380 (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 A062380 Cf. A000005, A000010, A060648, A062949.
%K A062380 nonn,mult
%O A062380 1,2
%A A062380 _Vladeta Jovovic_, Jul 07 2001