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

1,2

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 J. Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A140479 A264689 A146158 * A257644 A135940 A126430

Adjacent sequences: A062351 A062352 A062353 * A062355 A062356 A062357

Keyword

easy,nonn,mult

Author

Jason Earls, Jul 06 2001

Status

approved