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Lesser of twin primes p >= 3 for which sigma(p+1)/sigma(p-1) reaches record value, where sigma(n) is the divisor sum function (A000203).
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%I #29 Nov 01 2019 10:38:52

%S 3,7559,42839,55439,110879,415799,1713599,1940399,2489759,6652799,

%T 6846839,15855839,31600799,85765679,232792559,845404559,1470268799,

%U 6299092799,10708457759,17459441999,32125373279,135019684799,439977938399,449755225919,1799020903679,2126560035599,2835413380799,6278415343199

%N Lesser of twin primes p >= 3 for which sigma(p+1)/sigma(p-1) reaches record value, where sigma(n) is the divisor sum function (A000203).

%C Garcia et al. proved that assuming Dickson's conjecture, {sigma(p+1)/sigma(p-1) : p and p+2 are prime} is dense in [0, oo), and thus this sequence is infinite.

%H Stephan Ramon Garcia, Florian Luca, Kye Shi, Gabe Udell, <a href="https://arxiv.org/abs/1906.05927">Primitive root bias for twin primes II: Schinzel-type theorems for totient quotients and the sum-of-divisors function</a>, arXiv:1906.05927 [math.NT], 2019.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dickson%27s_conjecture">Dickson's conjecture</a>.

%e The values of sigma(p+1)/sigma(p-1) for the first terms are 2.333... < 2.539... < 2.621... < 2.734... < 2.836...

%t s = {}; rm = 0; p = 2; Do[q = NextPrime[p]; If[q - p != 2, p = q; Continue[]]; r = DivisorSigma[1, p + 1]/DivisorSigma[1, p - 1]; If[r > rm, rm = r; AppendTo[s, p]]; p = q, {10^6}]; s

%Y Cf. A000203, A001359, A006512, A008332, A008333, A326356.

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 11 2019

%E a(22)-a(28) from _Giovanni Resta_, Nov 01 2019