A298206 a(n) is the smallest b >= 2 such that b^(6*2^n) - b^(3*2^n) + 1 is prime.

6, 3, 3, 6, 5, 106, 207, 569, 224, 736, 2854, 21234, 14837, 165394, 24743, 62721, 237804, 143332

Offset: 0

Author

Serge Batalov, Jan 14 2018

Keywords

• nonn
• hard
• more

Comments

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

Examples

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

Links

• Phil Carmody, Prime Internet Eisenstein Search (ca. 2004-2005)
• Mersenneforum, Prime Internet Eisenstein Search discussion
• The Prime Pages, Generalized unique primes

Crossrefs

Subsequence of A205506.

Cf. A001620, A085398, A153438, A246119.

Programs

PARI

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

Formula

a(n) = A085398(18*2^n). - Jinyuan Wang, Dec 21 2022

Extensions

a(13) from Serge Batalov, Jan 24 2018