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

1,2

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).

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.

Example

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]

Crossrefs

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

Sequence in context: A048054 A284973 A239853 * A149170 A357447 A024053

Adjacent sequences: A209076 A209077 A209078 * A209080 A209081 A209082

Keyword

nonn

Author

Alexei Kourbatov, Mar 04 2012

Status

approved