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A209079 Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m. 2
1, 4, 9, 44, 96, 312, 2139, 4421, 48234, 623336, 1266781, 3897787, 20138571, 341171088, 6464294306, 148397712765, 299150944780, 8665061848812, 268337399189042, 1911903969221925, 5783509506896323, 213833540687410017 (list; graph; refs; listen; history; text; internal format)
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 hypoth├Ęse 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 A024053 A197970

Adjacent sequences:  A209076 A209077 A209078 * A209080 A209081 A209082

KEYWORD

nonn

AUTHOR

Alexei Kourbatov, Mar 04 2012

STATUS

approved

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Last modified April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)