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

1, 4, 9, 44, 96, 312, 2139, 4421, 48234, 623336, 1266781, 3897787, 20138571, 341171088, 6464294306, 148397712765, 299150944780, 8665061848812, 268337399189042, 1911903969221925, 5783509506896323, 213833540687410017

Offset: 1

Author

Alexei Kourbatov, Mar 04 2012

Keywords

• nonn

Comments

The sequence is increasing about as fast as the sequence of colossally abundant (CA) numbers (A004490).

We have two results:

(1) sigma(m)*phi(m)/m ~ m as m tends to infinity.

Here gamma is the Euler-Mascheroni constant 0.5772156649... (A001620).

Formula (1) follows from these known facts for CA numbers m:

(A) sigma(m)/m ~ exp(gamma) * log(log(m))

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

Dividing (A) by (B) we get sigma(m)*phi(m)/(m^2) ~ 1, hence (1) is true.

(2) 6m/(pi^2) < sigma(m)*phi(m)/m < m, which follows from Theorem 329 (Hardy and Wright, p. 352).

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).

Examples

			1 = [3*1/2]
4 = [12*2/6]
9 = [28*4/12]
44 = [168*16/60]
96 = [360*32/120]
312 = [1170*96/360]
2139 = [9360*576/2520]
4421 = [19344*1152/5040]
48234 = [232128*11520/55440]
623336 = [3249792*138240/720720]
1266781 = [6604416*276480/1441440]
3897787 = [20321280*829440/4324320]
20138571 = [104993280*4147200/21621600]
341171088 = [1889879040*66355200/367567200]
6464294306 = [37797580800*1194393600/6983776800]
148397712765 = [907141939200*26276659200/160626866400]
299150944780 = [1828682956800*52553318400/321253732800]
8665061848812 = [54860488704000*1471492915200/9316358251200]
268337399189042 = [1755535638528000*44144787456000/288807105787200]
1911903969221925 = [12508191424512000*309013512192000/2021649740510400]
5783509506896323 = [37837279059148800*927040536576000/6064949221531200]
		

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..382
• 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.
• 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.

Crossrefs

Cf. A004490 (colossally abundant numbers), A001620, A073751, A185339.