A062354 a(n) = sigma(n)*phi(n).

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Author

Jason Earls, Jul 06 2001

Keywords

• easy
• nonn
• mult

Comments

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

a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002

Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

References

• D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

Links

• T. D. Noe, Table of n, a(n) for n=1..10000
• Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
• J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Crossrefs

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Programs

Mathematica

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

PARI

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Formula

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002

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

6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014

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

Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020

a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025