This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347138 #37 Sep 22 2021 03:20:05
%S A347138 3,293,461,11867,90089
%N A347138 Numbers k such that (100^k + 1)/101 is prime.
%C A347138 These are the repunit primes in base -100. It is unusual to represent numbers in a negative base, but it follows the same formulation as any base: numbers are represented as a sum of powers in that base, i.e., a0*1 + a1*b^1 + a2*b^2 + a3*b^3 ... Since the base is negative, the terms will be alternating positive/negative. For repunits the coefficients are all ones so the sum reduces to 1 + b + b^2 + b^3 + ... + b^(k-1) = (b^k-1)/(b-1). Since b is negative and k is an odd prime, the sum equals (|b|^k+1)/(|b|+1). For k=3, the sum is 9901, which is prime. As with all repunits, we only need to PRP test the prime exponents. The factors of repunits base -100 will be of the form p=2*k*m+1 where m must be even, which is common for (negative) bases that are squares.
%H A347138 Paul Bourdelais, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;417ab0d6.0906">A Generalized Repunit Conjecture</a>
%e A347138 3 is a term since (100^3 + 1)/101 = 9901 is a prime.
%t A347138 Do[ If[ PrimeQ[ (100^n + 1)/101], Print[n]], {n, 0, 18000}]
%o A347138 (PARI) is(n)=isprime((100^n+1)/101)
%Y A347138 Cf. A309533, A309532, A237052, A229145, A057191, A057182, A057175.
%K A347138 nonn,hard,more
%O A347138 1,1
%A A347138 _Paul Bourdelais_, Aug 19 2021