cp's OEIS Frontend

A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).

%I A067698 #68 Jul 23 2025 09:09:23
%S A067698 2,3,4,5,6,8,9,10,12,16,18,20,24,30,36,48,60,72,84,120,180,240,360,
%T A067698 720,840,2520,5040
%N A067698 Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).
%C A067698 Previous name was: Numbers with relatively many and large divisors.
%C A067698 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.
%C A067698 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
%C A067698   lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).
%C A067698 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
%H A067698 Keith Briggs, <a href="https://projecteuclid.org/euclid.em/1175789744">Abundant numbers and the Riemann Hypothesis</a>, Experimental Math., Vol. 15, No. 2 (2006), p. 251-256.
%H A067698 Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/l33/l33.Abstract.html">Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis</a>, Integers 11 (2011), #A33.
%H A067698 Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, <a href="https://doi.org/10.1007/s11139-012-9371-0">On SA, CA, and GA numbers</a>, The Ramanujan Journal, Vol. 29 (2012), pp. 359-384; <a href="http://arxiv.org/abs/1112.6010">arXiv preprint</a>, arXiv:1112.6010 [math.NT], 2011-2012.
%H A067698 Jeffrey C. Lagarias, <a href="https://doi.org/10.1080/00029890.2002.11919883">An elementary problem equivalent to the Riemann hypothesis</a>, The American Mathematical Monthly, Vol. 109, No. 6 (2002), pp. 534-543; <a href="https://web.archive.org/web/20160406115509id_/http://web.maths.unsw.edu.au/~maikemassierer/teaching/zahlentheorie-fs12/LagariasRH.pdf">alternative copy</a>; <a href="https://arxiv.org/abs/math/0008177">arXiv preprint</a>, arXiv:math/0008177 [math.NT], 2000-2001.
%H A067698 Thomas Morrill and David John Platt, <a href="https://math.colgate.edu/~integers/v28/v28.pdf">Robin's inequality for 20-free integers</a>, Integers, Vol. 21 (2021), #A28.
%H A067698 Guy Robin, <a href="https://www.researchgate.net/publication/285597708_Robin1984">Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann</a>, J. Math. Pures Appl. 63 (1984), 187-213.
%H A067698 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GronwallsTheorem.html">Gronwall's Theorem</a>.
%H A067698 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RobinsTheorem.html">Robin's Theorem</a>.
%e A067698 9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
%p A067698 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:
%t A067698 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 *)
%t A067698 Select[Range[2,5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1,#]<0 &] (* _Ant King_, Feb 28 2013 *)
%o A067698 (PARI) is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ _Charles R Greathouse IV_, Feb 08 2017
%o A067698 (Python) from sympy import divisor_sigma, EulerGamma, E, log
%o A067698 print([n for n in range(2, 5041) if divisor_sigma(n) >= (E**EulerGamma * n * log(log(n)))]) # _Karl-Heinz Hofmann_, Apr 22 2022
%Y A067698 Cf. A057641 (based on Lagarias's extension of Robin's result).
%Y A067698 Cf. A000203, A004394, A073004, A091901, A189686, A196229.
%K A067698 nonn,nice
%O A067698 1,1
%A A067698 Ulrich Schimke (ulrschimke(AT)aol.com)
%E A067698 Edited by _N. J. A. Sloane_ at the suggestion of _Max Alekseyev_, Jul 17 2007
%E A067698 New name from _Jud McCranie_, Aug 14 2017