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 A164512 #13 Feb 16 2025 08:33:11
%S A164512 2,3,3,4,4,5,7,8,8,9,16,17,31,32,127,128,256,257,8191,8192,65536,
%T A164512 65537,131071,131072,524287,524288,2147483647,2147483648,
%U A164512 2305843009213693951,2305843009213693952,618970019642690137449562111,618970019642690137449562112
%N A164512 Prime power pairs of form (p^a, q^b = p^a + 1), a >= 1, b >= 1.
%C A164512 Consecutive prime powers with positive exponents.
%C A164512 a(n) = Ordered union of {2^3, 3^2, Fermat primes, Fermat primes - 1, Mersenne primes, Mersenne primes + 1}.
%C A164512 It is not known whether this sequence is infinite (but it is believed to be).
%C A164512 2^3, 3^2 are the only consecutive prime powers with exponents >= 2 (this is a consequence of the Catalan-Mihailescu theorem).
%C A164512 Only the first 5 Fermat numbers f_0 to f_4 are known to be prime.
%C A164512 It is conjectured that there exist an infinite number of Mersenne primes.
%H A164512 Daniel Forgues, <a href="/A164512/b164512.txt">Table of n, a(n) for n = 1..48</a>
%H A164512 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansConjecture.html">Catalan's Conjecture</a>.
%H A164512 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MersennePrime.html">Mersenne Prime</a>.
%H A164512 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatPrime.html">Fermat Prime</a>.
%Y A164512 Cf. A019434 (Fermat primes), A000668 (Mersenne primes).
%K A164512 hard,nonn
%O A164512 1,1
%A A164512 _Daniel Forgues_, Aug 14 2009
%E A164512 Edited by _N. J. A. Sloane_, Aug 24 2009