cp's OEIS Frontend

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

Consecutive prime powers.

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

k + (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 k such that k and k+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 -- Mihailescu 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.

Numbers k such that A003418(k) appears only once in the sequence A003418. This may suggest that k is also characterized by the pairs formed by a 2 whose direct neighbor is a prime number in the sequence A014963. - Eric Desbiaux, Feb 11 2015

The power graph and enhanced power graph of the groups PGL(2,q) have the same clique number iff q>1 is a term of this sequence (Peter Cameron's link). - Bernard Schott, Dec 14 2021

Numbers n such that n + (0, 4) is a prime power pair.

A generalization of the cousin primes. The cousin primes are a subsequence.

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 + (0, 3) is a prime power pair.

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.]

n + (0, 3) being a non-admissible pattern for prime pairs, has only prime power pairs (2^a - 3, 2^a) or (2^a, 2^a + 3), a >= 0.

Numbers n such that n and n+3 are primes would give only 2, for the prime pair (2, 5).

10^18 < a(28) <= 19807040628566084398385987581. - Donovan Johnson, Aug 17 2009

Numbers n such that n + (0, 5) is a prime power pair.

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+5 are primes would give only 2, for the prime pair (2, 7).

10^18 < a(21) <= 73786976294838206459. - Donovan Johnson, Aug 17 2009

Numbers n such that n + (0, 6) is a prime power pair.

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.]

At least one of a(n), a(n)+2, a(n)+4 and a(n)+6 must be a power of 3. See comments in A264734.

Intersection of A120431 and A164572.

a(14) > 10^3000, if it exists. Note that one among k, k+2, k+4 is always divisible by 3, so it must be a power of 3. - Giovanni Resta, Nov 12 2019