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Search: id:A094076
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| A094076 |
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Smallest k such that prime(n)+2^k is prime, or -1 if no such prime exists. |
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+0
9
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| 0, 1, 1, 2, 1, 2, 1, 2, 3, 1, 4, 2, 1, 2, 5, 3, 1, 8, 2, 1, 4, 2, 7, 3, 2, 1, 2, 1, 2, 7, 2, 3, 1, 10, 1, 4, 4, 2, 5, 3, 1, 4, 1, 2, 1, 6, 4, 2, 1, 2, 3, 1, 4, 5, 9, 3, 1, 20, 2, 1, 6, 7, 2, 1, 2, 5, 4, 4, 1, 2, 27, 3, 4, 4, 2, 15, 3, 2, 3, 10, 1, 8, 1, 4, 2, 7, 3, 2, 1, 2, 5, 3, 2, 3, 2, 7, 5, 1, 6, 4, 4, 9, 3, 1
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Conjecture: k > 0 for all n.
For all primes p < 1000 there exists a k such that p + 2^k is prime. However, for p = prime(321) = 2131, p + 2^k is not prime for all k < 30000. The conjecture may be in question. Similarly, I cannot find k such that p + 2^k is prime for p = 7013, 8543, 10711, 14033 for k < 20000. - Cino Hilliard (hillcino368(AT)gmail.com), Jun 27 2005
prime(80869739673507329) = 3367034409844073483, so a(80869739673507329) = -1 since 2^k + 3367034409844073483 is covered by {3, 5, 17, 257, 641, 65537, 6700417}. - Charles R Greathouse IV, Feb 08 2008
k=271129 is a smaller counterexample: gcd(k+2^n,2^24-1)>1 always holds using (1 mod 2, 0 mod 4, 2 mod 8, 6 mod 24, 14 mod 24 and 22 mod 24) as a covering for the n's. k with gcd(k+2^n,2^24-1)>1 always true were first found by Erdos (see refs). [From Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Mar 11 2009]
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REFERENCES
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P. Erdos, On integers of the form 2^k+p and some related problems, Summa Brasil. Math., 2 (1950), 113-123. [From Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Mar 11 2009]
A. O. L. Atkins and B. J. Birch, Computers in Number Theory, Academic Press, 1971, page 74. [From Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Mar 11 2009]
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LINKS
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Charles R Greathouse IV, Constructing a covering set for numbers 2^k + p
Charles R Greathouse IV, Table of n, a(n) for n = 1..3000 (with question marks at 321, 1066, 2168)
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EXAMPLE
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p = 773, k = 995, p + 2^k is prime.
p = 5101, k = 5760, p + 2^k is prime.
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PROGRAM
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(PARI) pplus2ton(n, m) = { local(k, s, p, y, flag); s=0; forprime(p=2, n, flag=1; for(k=0, m, y=p+2^k; if(ispseudoprime(y), print1(k, ", "); s++; flag=0; break) ); \ if(flag, print(p)); search for defiant primes. ); print(); print(s); } (Hilliard)
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CROSSREFS
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Cf. A067760.
Sequence in context: A083269 A097306 A102632 this_sequence A089611 A082067 A082061
Adjacent sequences: A094073 A094074 A094075 this_sequence A094077 A094078 A094079
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 29 2004
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EXTENSIONS
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More terms from Don Reble (djr(AT)nk.ca), May 2 2004
More terms from Cino Hilliard (hillcino368(AT)gmail.com), Jun 27 2005
More terms from Charles R Greathouse IV, Feb 08 2008
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