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%I #47 Aug 20 2020 03:19:16
%S 1,3,8,14,24,24,48,60,78,72,120,112,168,144,192,248,288,234,360,336,
%T 384,360,528,480,620,504,720,672,840,576,960,1008,960,864,1152,1092,
%U 1368,1080,1344,1440,1680,1152,1848,1680,1872,1584,2208,1984,2394,1860
%N a(n) = sigma(n)*phi(n).
%C Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
%C a(n) = Sum_{d|n} phi(n*d). - _Vladeta Jovovic_, Apr 17 2002
%C Apparently the Mobius transform of A062952. - _R. J. Mathar_, Oct 01 2011
%D D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.
%H T. D. Noe, <a href="/A062354/b062354.txt">Table of n, a(n) for n=1..10000</a>
%H Vaclav Kotesovec, <a href="/A062354/a062354.jpg">Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000</a>
%H J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1211.6944">Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis</a>, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.
%F Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - _Vladeta Jovovic_, Apr 17 2002
%F Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - _R. J. Mathar_, Oct 01 2011, corrected by _Vaclav Kotesovec_, Dec 17 2019
%F 6/Pi^2 < a(n)/n^2 < 1 for n > 1. - _Jonathan Sondow_, Mar 06 2014
%F Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - _Vaclav Kotesovec_, Dec 17 2019
%F Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - _Amiram Eldar_, Aug 20 2020
%t Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* _Carl Najafi_, Aug 16 2011 *)
%o (PARI) a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))
%Y Cf. A000010, A000203, A000252, A062355, A064840, A093827.
%K easy,nonn,mult
%O 1,2
%A _Jason Earls_, Jul 06 2001
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