A197639 GA2 numbers: n with G(n) >= G(a*n) for all integers a > 0, where G(k) = sigma(k)/(k*log(log(k))) and sigma(k) = sum of divisors of k.

3, 4, 5, 6, 8, 10, 12, 18, 24, 36, 48, 60, 72, 120, 180, 240, 360, 2520, 5040

Offset: 1

Author

Geoffrey Caveney, Jean-Louis Nicolas, and Jonathan Sondow, Dec 02 2011

Keywords

• nonn

Comments

Subsequence of A067698.

A member > 5040 exists iff the Riemann Hypothesis is false, in which case the sequence is infinite. In any case, 3 and 5 are the only odd members. (See Sections 1 and 4 of "On SA, CA, and GA numbers".)

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), article A33.
• 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. A000203, A067698, A197638, A201557.

Programs

Mathematica

nmax = 10^6; amax = 10;
G[k_] := DivisorSigma[1, k]/(k Log[Log[k]]);
okQ[n_] := DivisorSigma[1, n] > n Exp[EulerGamma] Log[Log[n]] && AllTrue[ Range[amax], Function[a, G[n] >= G[a*n]]];
Reap[For[n = 1, n <= nmax, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 30 2019 *)