A067698 - OEIS

A067698

Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040

COMMENTS

Previous name was: Numbers with relatively many and large divisors.

n is in the sequence iff sigma(n) >= exp(gamma) * n * log(log(n)), where gamma = Euler-Mascheroni constant and sigma(n) = sum of divisors of n.

Robin has shown that 5040 is the last element in the sequence iff the Riemann hypothesis is true. Moreover the sequence is infinite if the Riemann hypothesis is false. Gronwall's theorem says that

lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).

a(28) > 10^(10^13.11485), if the Riemann hypothesis is false (Morrill and Platt, 2021). Briggs (2006) found the lower bound 10^(10^10). - Amiram Eldar, Jul 23 2025

LINKS

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, On SA, CA, and GA numbers, The Ramanujan Journal, Vol. 29 (2012), pp. 359-384; arXiv preprint, arXiv:1112.6010 [math.NT], 2011-2012.

EXAMPLE

9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).

MAPLE

with (numtheory): expgam := exp(evalf(gamma)): for i from 2 to 6000 do: a := sigma (i): b := expgam*i*evalf(ln(ln(i))): if a >= b then print (i, a, b): fi: od:

MATHEMATICA

fQ[n_] := DivisorSigma[1, n] > n*Exp@ EulerGamma*Log@ Log@n; lst = {}; Do[ If[ fQ[n], AppendTo[lst, n]], {n, 2, 10^4}]; lst (* Robert G. Wilson v, May 16 2003 *)

Select[Range[2, 5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1, #]<0 &] (* Ant King, Feb 28 2013 *)

PROG

(PARI) is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ Charles R Greathouse IV, Feb 08 2017

(Python) from sympy import divisor_sigma, EulerGamma, E, log

print([n for n in range(2, 5041) if divisor_sigma(n) >= (E**EulerGamma * n * log(log(n)))]) # Karl-Heinz Hofmann, Apr 22 2022

CROSSREFS

Cf. A057641 (based on Lagarias's extension of Robin's result).

AUTHOR

Ulrich Schimke (ulrschimke(AT)aol.com)