1,2

Numbers n such that n + (0, 1) is a prime power pair.

Consecutive prime powers.

n + (0, 2m), m >= 1, being an admissible pattern for prime pairs, since (0, 2m) = (0, 0) (mod 2), has high density.

n + (0, 2m-1), m >= 1, being a non-admissible pattern for prime pairs, since (0, 2m-1) = (0, 1) (mod 2), has low density [the only possible pairs are (2^a - 2m-1, 2^a) or (2^a, 2^a + 2m-1), a >= 0.]

Numbers n such that n and n+1 are primes would give only 2, for the prime pair (2, 3).

This sequence corresponds to the least member of each one of the following prime power pairs, ordered by increasing value of least member: (1, 2), (2^3, 3^2), (Fermat primes - 1, Fermat primes), (Mersenne primes, Mersenne primes + 1}.

It is not known whether this sequence is infinite, but is conjectured to be since:

(*) 2^3, 3^2 are the only consecutive prime powers with exponents >= 2

(as a consequence of Mihailescu's theorem who proved Catalan's conjecture in 2002;)

(*) Only the first 5 Fermat numbers f_0 to f_4 are known to be prime

(it is conjectured that there might be no others, f_5 to f_32 are all composite);

(*) It has been conjectured that there exist an infinite number of Mersenne primes.

R. K. Guy, Unsolved Problems in Number Theory, D9.

P. Ribenboim, 13 Lect. on Fermat's Last Theorem, p. 236.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David W. Wilson and Eric Rains (rains(AT)caltech.edu) found a simple proof that in this case of Catalan's conjecture either n or n+1 must be a power of 2 and the other number must be a prime, except for n=8. Using this the sequence is easy to extend.

Daniel Forgues, Table of n, a(n) for n=1..25

Weisstein, Eric W., Catalan's Conjecture

Weisstein, Eric W., Mersenne Prime

Weisstein, Eric W., Fermat Prime

Do[ a = Length[ FactorInteger[ 2^n - 1 ] ]; b = Length[ FactorInteger[ 2^n ] ]; c = Length[ FactorInteger[ 2^n + 1 ] ]; If[ a == b, Print[ 2^n - 1 ] ]; If[ b == c, Print[ 2^n ] ], {n, 0, 127} ]

Cf. A000961.

Cf. A019434 Fermat primes: primes of form 2^(2^n) + 1, n >= 0.

Cf. A000668 Mersenne primes (of form 2^p - 1 where p is a prime).

Cf. A120431 Numbers n such that n and n+2 are prime powers.

Cf. A164571 Numbers n such that n and n+3 are prime powers.

Cf. A164572 Numbers n such that n and n+4 are prime powers.

Cf. A164573 Numbers n such that n and n+5 are prime powers.

Cf. A164574 Numbers n such that n and n+6 are prime powers.

Sequence in context: A126850 A007497 A126882 this_sequence A159554 A101128 A119345

Adjacent sequences: A006546 A006547 A006548 this_sequence A006550 A006551 A006552

nonn,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from David W. Wilson (davidwwilson(AT)comcast.net)

Additional comments from Daniel Forgues (squid(AT)zensearch.com), Aug 17 2009