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A185339 Integer part of sigma(m)/phi(m) for colossally abundant numbers m. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A310134 A039591 A114082 * A189015 A189018 A189133

Adjacent sequences:  A185336 A185337 A185338 * A185340 A185341 A185342

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Feb 28 2012

STATUS

approved

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Last modified March 29 00:58 EDT 2020. Contains 333104 sequences. (Running on oeis4.)