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

1,1

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.

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

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.

Example

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]

Crossrefs

Cf. A004490 (colossally abundant numbers), A073751.

Sequence in context: A039591 A114082 A343218 * A189015 A189018 A189133

Adjacent sequences: A185336 A185337 A185338 * A185340 A185341 A185342

Keyword

nonn

Author

Alexei Kourbatov, Feb 28 2012

Status

approved