A185339 Integer part of sigma(m)/phi(m) for colossally abundant numbers m.

3, 6, 7, 10, 11, 12, 16, 16, 20, 23, 23, 24, 25, 28, 31, 34, 34, 37, 39, 40, 40, 43, 45, 47, 47, 49, 51, 53, 53, 55, 57, 58, 60, 60, 62, 62, 64, 64, 65, 67, 67, 68, 70, 71, 72, 74, 75, 76, 77, 78, 78, 79, 80, 81, 82, 83, 84, 84, 85, 86, 87, 88, 89, 90, 90

Offset: 1

Author

Alexei Kourbatov, Feb 28 2012

Keywords

• nonn

Comments

The sequence is nondecreasing - this follows from the properties of the sum-of-divisors (sigma) and Euler's totient (phi) functions. Many terms appear more than once. Each integer greater than 73 appears at least once.

Colossally abundant (CA) numbers m are listed in A004490.

Examples

			3 = [3/1]     for m=2: sigma(2)=3 and phi(2)=1;
6 = [12/2]    for m=6: sigma(6)=12 and phi(6)=2;
7 = [28/4]    for m=12: sigma(12)=28 and phi(12)=4;
10 = [168/16] for m=60 (see A004490 for further values of m);
11 = [360/32]
12 = [1170/96]
16 = [9360/576]
16 = [19344/1152]
20 = [232128/11520]
23 = [3249792/138240]
23 = [6604416/276480]
24 = [20321280/829440]
25 = [104993280/4147200]
28 = [1889879040/66355200]
31 = [37797580800/1194393600]
34 = [907141939200/26276659200]
34 = [1828682956800/52553318400]
37 = [54860488704000/1471492915200]
39 = [1755535638528000/44144787456000]
40 = [12508191424512000/309013512192000]
40 = [37837279059148800/927040536576000]
43 = [1437816604247654400/33373459316736000]
		

References

• G. H. Hardy and E.M. Wright, An introduction to the theory of numbers, 6th edition, Oxford University Press (2008), 350-353.
• G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63 (1984), 187-213.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• L. Alaoglu and P. Erdos, On highly composite and similar numbers, Trans. Amer. Math. Soc., 56 (1944), 448-469. Errata
• Keith Briggs, Abundant numbers and the Riemann Hypothesis, Experimental Math., Vol. 16 (2006), p. 251-256.
• T. H. Grönwall, Some asymptotic expressions in the theory of numbers, Trans. Amer. Math. Soc 14 (1913), 113-122.
• J.-L. Nicolas, Petites valeurs de la fonction d'Euler, J. Number Theory 17, no.3 (1983), 375-388.
• S. Ramanujan, Highly composite numbers, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
• Eric W. Weisstein, MathWorld: Robin's Theorem

Crossrefs

Cf. A004490 (colossally abundant numbers), A073751.

Formula

(1) sigma(m)/phi(m) ~ exp(2*gamma)*(log(log(m)))^2 as m tends to infinity.

Here gamma is the Euler constant, gamma = 0.5772156649...

Formula (1) can be proved based on two known facts for CA numbers m:

(A) sigma(m)/m ~ exp(gamma) * log(log(m)) [see Ramanujan, 1997, eq. 383]

(B) m/phi(m) ~ exp(gamma) * log(log(m))

(we get (1) simply by multiplying (A) and (B) together).

The following empirical inequality suggests that sigma(m)/phi(m) approximates the limiting sequence exp(2*gamma)*(log(log(m)))^2 from below:

(2) sigma(m)/phi(m) < exp(2*gamma)*(log(log(m)))^2 for large enough CA numbers m (namely, for m>10^35, i.e., beginning with the 34th CA number m). No formal proof is known for formula (2). If a proof of (2) becomes available, then Robin's inequality sigma(m)/m < exp(gamma) * log(log(m)) (and therefore the Riemann Hypothesis) will follow as well. Thus (2) must be exceedingly difficult to prove.