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%I #55 May 05 2022 15:08:12
%S 2,3,4,5,6,8,9,10,12,16,18,20,24,30,36,48,60,72,84,120,180,240,360,
%T 720,840,2520,5040
%N Positive integers such that sigma(n) >= exp(gamma) * n * log(log(n)).
%C Previous name was: Numbers with relatively many and large divisors.
%C 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 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 lim sup_{n -> infinity} sigma(n)/(n*log(log(n))) = exp(gamma).
%D Guy Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.
%H G. Caveney, J.-L. Nicolas, and J. 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 G. Caveney, J.-L. Nicolas and J. Sondow, <a href="http://arxiv.org/abs/1112.6010">On SA, CA, and GA numbers</a>, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.
%H J. C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GronwallsTheorem.html">Gronwall's Theorem</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RobinsTheorem.html">Robin's Theorem</a>
%e 9 is in the sequence since sigma(9) = 13 > 12.6184... = exp(gamma) * 9 * log(log(9)).
%p 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 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 Select[Range[2,5050], Exp[EulerGamma] # Log[Log[#]]-DivisorSigma[1,#]<0 &] (* _Ant King_, Feb 28 2013 *)
%o (PARI) is(n)=sigma(n) >= exp(Euler) * n * log(log(n)) \\ _Charles R Greathouse IV_, Feb 08 2017
%o (Python) from sympy import divisor_sigma, EulerGamma, E, log
%o 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 Cf. A057641 (based on Lagarias' extension of Robin's result).
%Y Cf. A091901, A189686, A004394, A196229.
%K nonn,nice
%O 1,1
%A Ulrich Schimke (ulrschimke(AT)aol.com)
%E Edited by _N. J. A. Sloane_ at the suggestion of _Max Alekseyev_, Jul 17 2007
%E New name from _Jud McCranie_, Aug 14 2017
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