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%I #12 Dec 10 2021 11:35:27
%S 3,5,701,829,1031,1033,7229,19463,370421
%N Numbers k such that (81^k + 1)/82 is prime.
%C These are the Repunits in base -81. Since 81=3^4, factors will be of the form p=8nk+1. (Negative) bases that are powers of small numbers appear to have a higher frequency of primes than Repunits in other bases. The best linear fit for this base is currently 0.29918 which is much lower (better) than the conjectured 0.56145948 (see link to conjecture).
%H Paul Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>
%e 3 is a term since (81^3 + 1)/82 = 6481 is a prime.
%t Do[ If[ PrimeQ[ (81^n+1)/82], Print[n]], {n, 0, 1000000}]
%o (PARI) is(n)=isprime((81^n+1)/82)
%Y Cf. A309533, A309532, A237052, A229145, A057191, A057182, A057175.
%K nonn,hard,more
%O 1,1
%A _Paul Bourdelais_, Dec 09 2021