A326393 Primes p for which sigma(p+1)/sigma(p) reaches a record value, where sigma(k) is the divisor sum function (A000203).

2, 3, 5, 11, 23, 47, 59, 167, 179, 239, 359, 719, 839, 1259, 3359, 5039, 10079, 35279, 37799, 55439, 110879, 166319, 665279, 831599, 1081079, 1441439, 6320159, 6486479, 12972959, 24504479, 61261199, 82162079, 136936799, 232792559, 410810399, 698377679, 735134399

Offset: 1

Author

Amiram Eldar, Sep 11 2019

Keywords

• nonn

Comments

Garcia et al. proved that {sigma(p+1)/sigma(p) : p prime} is dense in [3/2, oo), and thus this sequence is infinite.

Examples

			The values of sigma(p+1)/sigma(p) for the first terms are 1.333... < 1.75 < 2 < 2.333... < 2.5 < ...
		

Links

• Stephan Ramon Garcia, Florian Luca, Kye Shi, and Gabe Udell, Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function, arXiv:1906.05927 [math.NT], 2019.

Crossrefs

Cf. A000203, A008333.

Programs

Mathematica

s = {}; rm = 0; p = 2; Do[q = NextPrime[p]; r = DivisorSigma[1, p + 1]/DivisorSigma[1, p]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^3}]; s