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Primes of the form 54k+1 generated recursively. Initial prime is 109. General term is a(n) = Min {p is prime; p divides (R^27 - 1)/(R^9 - 1); p == 1 (mod 27)}, where Q is the product of previous terms in the sequence and R = 3*Q.
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%I #16 Feb 11 2024 14:19:23

%S 109,50221,379,5077,2527181639419400128997560106426867837203,112807,

%T 2094067,1567,9325207,370603,67447,27978113462777647321591,1012771,

%U 163,396577,7096357,3511,3673,541,389287,1999,68979565009,649108891

%N Primes of the form 54k+1 generated recursively. Initial prime is 109. General term is a(n) = Min {p is prime; p divides (R^27 - 1)/(R^9 - 1); p == 1 (mod 27)}, where Q is the product of previous terms in the sequence and R = 3*Q.

%C All prime divisors of (R^27 - 1)/(R^9 - 1) different from 3 are congruent to 1 modulo 54.

%D M. Ram Murty, Problems in Analytic Number Theory, Springer-Verlag, NY, (2001), p. 209.

%e a(2) = 50221 is the smallest prime divisor congruent to 1 mod 54 of

%e (R^27-1)/(R^9- 1) = 1827509098737085519727094436535854935801097657 = 50221 * 106219 * 342587871163695447795790279515751543, where Q = 109 and R = 3*Q.

%Y Cf. A000945, A057204-A057208, A051308-A051335, A124984-A124993, A125037-A125045.

%K nonn

%O 1,1

%A _Nick Hobson_, Nov 18 2006

%E More terms from _Sean A. Irvine_, Dec 11 2011