A353076 Odd positive integers k such that sigma(k) > exp(gamma) * k * log(log(k))/2.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 81, 87, 93, 99, 105, 117, 135, 147, 153, 165, 171, 189, 195, 207, 225, 231, 255, 273, 285, 297, 315, 345, 351, 357, 375, 399, 405, 435, 441, 465, 495, 525, 555, 567, 585

Offset: 1

Author

Amiram Eldar, Apr 22 2022

Keywords

• nonn

Comments

The first 23 oddly colossally abundant numbers (A110464) are in this sequence.

According to a proof by Washington and Yang (2021), the Riemann hypothesis is equivalent to the statement that all the terms of this sequence are smaller than A110464(24) = 18565284664427130919514350125.

Examples

			3 is in the sequence since 3 is odd and sigma(3) = 4 > exp(gamma) * 3 * log(log(3))/2 = 0.251... .
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..470
• Lawrence C. Washington and Ambrose Yang, Analogues of the Robin-Lagarias criteria for the Riemann hypothesis, International Journal of Number Theory, Vol. 17, No. 4 (2021), pp. 843-870; arXiv preprint, arXiv:2008.04787 [math.NT], 2020.

Crossrefs

Cf. A000203 (sigma), A067698, A073004 (exp(gamma)), A110464.

Programs

Mathematica

Select[Range[3, 600, 2], DivisorSigma[1, #] > Exp[EulerGamma] * # * Log[Log[#]]/2 &]

Python

from sympy import divisor_sigma, EulerGamma, E, log
print([k for k in range(3, 600, 2) if divisor_sigma(k) > (E**EulerGamma * k * log(log(k)) / 2)]) # Karl-Heinz Hofmann, Apr 22 2022