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A006549
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Numbers k such that k and k+1 are prime powers.
(Formerly M0582)
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23
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1, 2, 3, 4, 7, 8, 16, 31, 127, 256, 8191, 65536, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727
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listen;
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
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LINKS
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MATHEMATICA
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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} ]
Join[{1}, SequencePosition[Boole[PrimePowerQ[Range[600000]]], {1, 1}][[All, 1]]] (* Requires Mathematica version 10 or later *) (* Generates the first 14 terms of the sequence. Increase Range constant to generate more. *) (* Harvey P. Dale, Apr 12 2020 *)
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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