A298206 - OEIS

%I #29 Apr 03 2023 10:36:13

%S 6,3,3,6,5,106,207,569,224,736,2854,21234,14837,165394,24743,62721,

%T 237804,143332

%N a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.

%C a(13) = 165394 is a significant outlier from the generally expected trend, which can be conjectured to be 6*2^n*gamma, where gamma is the Euler-Mascheroni constant A001620. Additionally, the next b > a(13) such that b^(6*2^n) - b^(3*2^n) + 1 is prime is 165836, which is remarkably close to a(13). - _Serge Batalov_, Jan 24 2018

%H Phil Carmody, <a href="http://fatphil.org/maths/PIES/">Prime Internet Eisenstein Search</a> (ca. 2004-2005)

%H Mersenneforum, <a href="http://mersenneforum.org/showthread.php?t=19655">Prime Internet Eisenstein Search discussion</a>

%H The Prime Pages, <a href="https://t5k.org/top20/page.php?id=44">Generalized unique primes</a>

%F a(n) = A085398(18*2^n). - _Jinyuan Wang_, Dec 21 2022

%e 2^12 - 2^6 + 1 = 4033 is composite and 3^12 - 3^6 + 1 = 530713 is prime, so a(1) = 3.

%o (PARI) for(n=0,9,for(b=2,1000,x=b^(3*2^n); if(isprime(x*(x-1)+1), print1(b,", "); break)))

%Y Subsequence of A205506.

%Y Cf. A001620, A085398, A153438, A246119.

%K nonn,hard,more

%O 0,1

%A _Serge Batalov_, Jan 14 2018

%E a(13) from _Serge Batalov_, Jan 24 2018