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A006549 Numbers n such that n and n+1 are prime powers.
(Formerly M0582)
22
1, 2, 3, 4, 7, 8, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 -- 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 n such that A003418(n) appears only once in the sequence A003418. This may suggest that n 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

REFERENCES

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.

LINKS

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

Eric Weisstein's World of Mathematics, Catalan's Conjecture

Eric Weisstein's World of Mathematics, Mersenne Prime

Eric Weisstein's World of Mathematics, Fermat Prime

MATHEMATICA

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} ]

PROG

(Haskell)

a006549 n = a006549_list !! (n-1)

a006549_list = [1, 2, 3, 4, 7, 8] ++ f (drop 4 a000040_list) where

   f (p:ps) | a010055 (p - 1) == 1 = (p - 1) : f ps

            | a010055 (p + 1) == 1 = p : f ps

            | otherwise            = f ps

-- Reinhard Zumkeller, Jan 03 2013

(PARI) is(n)=if(n<5, return(n>0)); isprimepower(n) && isprimepower(n+1) \\ Charles R Greathouse IV, Apr 24 2015

CROSSREFS

Cf. A000961, A000040, A010055.

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: A126882 A239973 A281782 * A134459 A225211 A159554

Adjacent sequences:  A006546 A006547 A006548 * A006550 A006551 A006552

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

Additional comments from Daniel Forgues, Aug 17 2009

STATUS

approved

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Last modified March 19 15:02 EDT 2019. Contains 321330 sequences. (Running on oeis4.)