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A209079 Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m.

%I A209079 #39 Aug 20 2019 09:25:06
%S A209079 1,4,9,44,96,312,2139,4421,48234,623336,1266781,3897787,20138571,
%T A209079 341171088,6464294306,148397712765,299150944780,8665061848812,
%U A209079 268337399189042,1911903969221925,5783509506896323,213833540687410017
%N A209079 Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m.
%C A209079 The sequence is increasing about as fast as the sequence of colossally abundant (CA) numbers (A004490).
%C A209079 We have two results:
%C A209079 (1) sigma(m)*phi(m)/m ~ m  as m tends to infinity.
%C A209079 Here gamma is the Euler-Mascheroni constant 0.5772156649... (A001620).
%C A209079 Formula (1) follows from these known facts for CA numbers m:
%C A209079   (A) sigma(m)/m ~ exp(gamma) * log(log(m))
%C A209079   (B)   m/phi(m) ~ exp(gamma) * log(log(m))
%C A209079 Dividing (A) by (B) we get sigma(m)*phi(m)/(m^2) ~ 1, hence (1) is true.
%C A209079 (2) 6m/(pi^2) < sigma(m)*phi(m)/m < m, which follows from Theorem 329 (Hardy and Wright, p. 352).
%C A209079 Ramanujan was the first to establish (A) for CA numbers m (see equation 383 in Ramanujan's paper; note that he used a different name for CA numbers: generalized superior highly composite numbers). Once we have (A) for an increasing sequence of numbers m (including, but not limited to CA numbers m), then (B) easily follows from (A) because, for large m, sigma(m)/m < m/phi(m) < exp(gamma) log(log(m)) + 0.6/(log(log(m))) (see Robin, 1984, p. 206).
%D A209079 G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th edition, Oxford University Press (2008), 350-353.
%D A209079 G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. J. Math. Pures Appl. 63 (1984), 187-213.
%H A209079 Amiram Eldar, <a href="/A209079/b209079.txt">Table of n, a(n) for n = 1..382</a>
%H A209079 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 A209079 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 A209079 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 A209079 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.
%e A209079 1 = [3*1/2]
%e A209079 4 = [12*2/6]
%e A209079 9 = [28*4/12]
%e A209079 44 = [168*16/60]
%e A209079 96 = [360*32/120]
%e A209079 312 = [1170*96/360]
%e A209079 2139 = [9360*576/2520]
%e A209079 4421 = [19344*1152/5040]
%e A209079 48234 = [232128*11520/55440]
%e A209079 623336 = [3249792*138240/720720]
%e A209079 1266781 = [6604416*276480/1441440]
%e A209079 3897787 = [20321280*829440/4324320]
%e A209079 20138571 = [104993280*4147200/21621600]
%e A209079 341171088 = [1889879040*66355200/367567200]
%e A209079 6464294306 = [37797580800*1194393600/6983776800]
%e A209079 148397712765 = [907141939200*26276659200/160626866400]
%e A209079 299150944780 = [1828682956800*52553318400/321253732800]
%e A209079 8665061848812 = [54860488704000*1471492915200/9316358251200]
%e A209079 268337399189042 = [1755535638528000*44144787456000/288807105787200]
%e A209079 1911903969221925 = [12508191424512000*309013512192000/2021649740510400]
%e A209079 5783509506896323 = [37837279059148800*927040536576000/6064949221531200]
%Y A209079 Cf. A004490 (colossally abundant numbers), A001620, A073751, A185339.
%K A209079 nonn
%O A209079 1,2
%A A209079 _Alexei Kourbatov_, Mar 04 2012