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

%I A185339 #58 Feb 16 2025 08:33:13
%S A185339 3,6,7,10,11,12,16,16,20,23,23,24,25,28,31,34,34,37,39,40,40,43,45,47,
%T A185339 47,49,51,53,53,55,57,58,60,60,62,62,64,64,65,67,67,68,70,71,72,74,75,
%U A185339 76,77,78,78,79,80,81,82,83,84,84,85,86,87,88,89,90,90
%N A185339 Integer part of sigma(m)/phi(m) for colossally abundant numbers m.
%C A185339 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.
%C A185339 Colossally abundant (CA) numbers m are listed in A004490.
%D A185339 G. H. Hardy and E.M. Wright, An introduction to the theory of numbers, 6th edition, Oxford University Press (2008), 350-353.
%D A185339 G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63 (1984), 187-213.
%H A185339 Amiram Eldar, <a href="/A185339/b185339.txt">Table of n, a(n) for n = 1..10000</a>
%H A185339 L. Alaoglu and P. Erdos, <a href="http://www.renyi.hu/~p_erdos/1944-03.pdf">On highly composite and similar numbers,</a> Trans. Amer. Math. Soc., 56 (1944), 448-469. <a href="http://upforthecount.com/math/errata.html">Errata</a>
%H A185339 Keith Briggs, <a href="http://projecteuclid.org/euclid.em/1175789744">Abundant numbers and the Riemann Hypothesis</a>, Experimental Math., Vol. 16 (2006), p. 251-256.
%H A185339 T. H. Grönwall, <a href="http://dx.doi.org/10.1090/S0002-9947-1913-1500940-6">Some asymptotic expressions in the theory of numbers</a>, Trans. Amer. Math. Soc 14 (1913), 113-122.
%H A185339 J.-L. Nicolas, <a href="http://dx.doi.org/10.1016/0022-314X(83)90055-0">Petites valeurs de la fonction d'Euler</a>, J. Number Theory 17, no.3 (1983), 375-388.
%H A185339 S. Ramanujan, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly composite numbers</a>, Annotated and with a foreword by J.-L. Nicolas and G. Robin, Ramanujan J., 1 (1997), 119-153.
%H A185339 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/RobinsTheorem.html">MathWorld: Robin's Theorem</a>
%F A185339 (1) sigma(m)/phi(m) ~ exp(2*gamma)*(log(log(m)))^2 as m tends to infinity.
%F A185339 Here gamma is the Euler constant, gamma = 0.5772156649...
%F A185339 Formula (1) can be proved based on two known facts for CA numbers m:
%F A185339 (A) sigma(m)/m ~ exp(gamma) * log(log(m)) [see Ramanujan, 1997, eq. 383]
%F A185339 (B) m/phi(m) ~ exp(gamma) * log(log(m))
%F A185339 (we get (1) simply by multiplying (A) and (B) together).
%F A185339 The following empirical inequality suggests that sigma(m)/phi(m) approximates the limiting sequence exp(2*gamma)*(log(log(m)))^2 from below:
%F A185339 (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.
%e A185339 3 = [3/1]     for m=2: sigma(2)=3 and phi(2)=1;
%e A185339 6 = [12/2]    for m=6: sigma(6)=12 and phi(6)=2;
%e A185339 7 = [28/4]    for m=12: sigma(12)=28 and phi(12)=4;
%e A185339 10 = [168/16] for m=60 (see A004490 for further values of m);
%e A185339 11 = [360/32]
%e A185339 12 = [1170/96]
%e A185339 16 = [9360/576]
%e A185339 16 = [19344/1152]
%e A185339 20 = [232128/11520]
%e A185339 23 = [3249792/138240]
%e A185339 23 = [6604416/276480]
%e A185339 24 = [20321280/829440]
%e A185339 25 = [104993280/4147200]
%e A185339 28 = [1889879040/66355200]
%e A185339 31 = [37797580800/1194393600]
%e A185339 34 = [907141939200/26276659200]
%e A185339 34 = [1828682956800/52553318400]
%e A185339 37 = [54860488704000/1471492915200]
%e A185339 39 = [1755535638528000/44144787456000]
%e A185339 40 = [12508191424512000/309013512192000]
%e A185339 40 = [37837279059148800/927040536576000]
%e A185339 43 = [1437816604247654400/33373459316736000]
%Y A185339 Cf. A004490 (colossally abundant numbers), A073751.
%K A185339 nonn
%O A185339 1,1
%A A185339 _Alexei Kourbatov_, Feb 28 2012