A189686 Superabundant numbers (A004394) satisfying the reverse of Robin's inequality (A091901).

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

Offset: 1

Author

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, May 30 2011

Keywords

• nonn

Comments

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

Links

• G. Caveney, J.-L. Nicolas, and J. Sondow, Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis, Integers 11 (2011), #A33 (see Table 1).
• G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384.

Crossrefs

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

Programs

Mathematica

kmax = 10^4;
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]]];
A067698 = Select[Range[2, kmax], DivisorSigma[1, #] > Exp[EulerGamma] # Log[Log[#]]&];
Intersection[A004394, A067698] (* Jean-François Alcover, Jan 28 2019 *)

PARI

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

Formula

Equals A004394 intersect A067698.

Extensions

Erroneous terms 1260 and 1680 removed by Jean-François Alcover, Jan 28 2019