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%I #25 Apr 03 2023 10:36:13
%S 2,3,88,28,688,7003,1925
%N Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.
%C Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=6^n.
%C Numbers of this form are Generalized unique primes. a(6) generates a 306477-digit prime.
%H C. Caldwell, <a href="https://t5k.org/top20/page.php?id=44">Generalized unique primes</a>
%F a(n) = A085398(6^(n+1)). - _Jinyuan Wang_, Jan 01 2023
%e When k = 88, k^72 - k^36 + 1 is prime. Since this isn't prime for k < 88, a(2) = 88.
%o (PARI) a(n)=k=1; while(!ispseudoprime(k^(6^n)*(k^(6^n)-1)+1), k++); k
%o n=0; while(n<100, print1(a(n), ", "); n++)
%Y Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246120.
%K nonn,more,hard
%O 0,1
%A _Serge Batalov_, Aug 14 2014
%E a(6) from _Serge Batalov_, Aug 15 2014
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