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Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).
4

%I #27 Jan 29 2019 04:36:16

%S 2,4,6,12,24,36,48,60,120,180,240,360,720,840,2520,5040

%N Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

%C 5040 is the last element in the sequence if and only if the Riemann Hypothesis is true. (See Akbary and Friggstad in A004394.)

%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 (see Table 1).

%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.

%F Equals A004394 intersect A067698.

%t kmax = 10^4;

%t A004394 = Join[{1}, Reap[For[r = 1; k = 2, k <= kmax, k = k + 2, s = DivisorSigma[-1, k]; If[s > r, r = s; Sow[k]]]][[2, 1]]];

%t A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];

%t Intersection[A004394, A067698] (* _Jean-François Alcover_, Jan 28 2019 *)

%o (PARI) is(n)=sigma(n) >= exp(Euler) * n * log(log(n)); \\ A067698

%o lista(nn) = my(r=1, t); forstep(n=2, nn, 2, t=sigma(n, -1); if(t>r && is(n), r=t; print1(n, ", "))); \\ _Michel Marcus_, Jan 28 2019; adapted from A004394

%Y Cf. A004394, A091901, A067698, A166981, A077006.

%K nonn

%O 1,1

%A Geoffrey Caveney, Jean-Louis Nicolas, and _Jonathan Sondow_, May 30 2011

%E Erroneous terms 1260 and 1680 removed by _Jean-François Alcover_, Jan 28 2019