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

1, 2, 8, 12, 32, 16, 72, 64, 108, 64, 200, 96, 288, 144, 256, 320, 512, 216, 648, 384, 576, 400, 968, 512, 1200, 576, 1296, 864, 1568, 512, 1800, 1536, 1600, 1024, 2304, 1296, 2592, 1296, 2304, 2048, 3200, 1152, 3528, 2400, 3456, 1936, 4232, 2560, 5292, 2400

Offset: 1

Author

Jonathan Vos Post, May 27 2007

Keywords

• easy
• nonn
• mult

Links

• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000005, A000010, A064840, A127473, A256392.

Programs

Magma

[ EulerPhi(n)*EulerPhi(n)*NumberOfDivisors(n) : n in [1..100] ];

Mathematica

Table[EulerPhi[n]^2 DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Dec 05 2012 *)

Formula

Multiplicative with a(p^e) = (e+1)*(p-1)^2*p^(2*e-2). - Amiram Eldar, Dec 29 2022

From Vaclav Kotesovec, May 31 2024: (Start)

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).

Let f(s) = Product_{p prime} (1 - 1/p^(2*s-2) + 2/p^(2*s-3) - 4/p^(s-1) + 2/p^s).

Sum_{k=1..n} a(k) ~ f(3) * n^3 * (log(n) + 2*gamma - 1/3 + f'(3)/f(3)) / 3, where

f(3) = Product_{p prime} (1 - 4/p^2 + 4/p^3 - 1/p^4) = A256392 = 0.2177787166195363783230075141194468131307977550013559376482764035236264...,

f'(3) = f(3) * Sum_{p prime} 2*(2*p - 1) * log(p) / (p^3 + p^2 - 3*p + 1) = f(3) * 1.6860441157206199528397247528679297282000614932962665074593283751342385...

and gamma is the Euler-Mascheroni constant A001620. (End)