cp's OEIS Frontend

A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).

%I A126775 #18 May 31 2024 06:45:16
%S A126775 1,2,8,12,32,16,72,64,108,64,200,96,288,144,256,320,512,216,648,384,
%T A126775 576,400,968,512,1200,576,1296,864,1568,512,1800,1536,1600,1024,2304,
%U A126775 1296,2592,1296,2304,2048,3200,1152,3528,2400,3456,1936,4232,2560,5292,2400
%N A126775 a(n) = phi(n)^2 * d(n) = A000010(n)^2 * A000005(n).
%H A126775 Vaclav Kotesovec, <a href="/A126775/a126775.jpg">Graph - the asymptotic ratio (100000 terms)</a>
%F A126775 Multiplicative with a(p^e) = (e+1)*(p-1)^2*p^(2*e-2). - _Amiram Eldar_, Dec 29 2022
%F A126775 From _Vaclav Kotesovec_, May 31 2024: (Start)
%F A126775 Dirichlet g.f.: zeta(s-2)^2 * Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
%F A126775 Let f(s) = Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).
%F A126775 Sum_{k=1..n} a(k) ~ f(3) * n^3 * (log(n) + 2*gamma - 1/3 + f'(3)/f(3)) / 3, where
%F A126775 f(3) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264...,
%F A126775 f'(3) = f(3) * Sum_{p prime} 2*(2*p - 1) * log(p) / (p^3 + p^2 - 3*p + 1) = f(3) * 1.6860441157206199528397247528679297282000614932962665074593283751342385...
%F A126775 and gamma is the Euler-Mascheroni constant A001620. (End)
%t A126775 Table[EulerPhi[n]^2 DivisorSigma[0,n],{n,50}] (* _Harvey P. Dale_, Dec 05 2012 *)
%o A126775 (Magma) [ EulerPhi(n)*EulerPhi(n)*NumberOfDivisors(n) : n in [1..100] ];
%Y A126775 Cf. A000005, A000010, A064840, A127473, A256392.
%K A126775 easy,nonn,mult
%O A126775 1,2
%A A126775 _Jonathan Vos Post_, May 27 2007