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%I #12 Feb 13 2024 02:58:50
%S 2,3,3,4,4,5,7,8,8,9,16,17,31,32,127,128,256,257,8191,8192,65536,
%T 65537,131071,131072,524287,524288,2147483647,2147483648,
%U 2305843009213693951,2305843009213693952,618970019642690137449562111,618970019642690137449562112
%N Prime power pairs of form (p^a, q^b = p^a + 1), a >= 1, b >= 1.
%C Consecutive prime powers with positive exponents.
%C a(n) = Ordered union of {2^3, 3^2, Fermat primes, Fermat primes - 1, Mersenne primes, Mersenne primes + 1}.
%C It is not known whether this sequence is infinite (but it is believed to be).
%C 2^3, 3^2 are the only consecutive prime powers with exponents >= 2 (this is a consequence of the Catalan-Mihailescu theorem).
%C Only the first 5 Fermat numbers f_0 to f_4 are known to be prime.
%C It is conjectured that there exist an infinite number of Mersenne primes.
%H Daniel Forgues, <a href="/A164512/b164512.txt">Table of n, a(n) for n = 1..48</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CatalansConjecture.html">Catalan's Conjecture</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MersennePrime.html">Mersenne Prime</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatPrime.html">Fermat Prime</a>.
%Y Cf. A019434 (Fermat primes), A000668 (Mersenne primes).
%K hard,nonn
%O 1,1
%A _Daniel Forgues_, Aug 14 2009
%E Edited by _N. J. A. Sloane_, Aug 24 2009
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