A094076 Smallest k such that prime(n) + 2^k is prime, or -1 if no such prime exists.

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

Offset

1,4

Comments

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, 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). - Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Mar 11 2009

References

A. O. L. Atkin and B. J. Birch, eds., Computers in Number Theory, Academic Press, 1971, page 74.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..7771 (terms 1..3000 from Charles R Greathouse IV).

P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123. [From Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Mar 11 2009]

Charles R Greathouse IV, Constructing a covering set for numbers 2^k + p [Cached copy]

Example

p = 773, k = 995, p + 2^k is prime.
p = 5101, k = 5760, p + 2^k is prime.

Mathematica

sk[n_]:=Module[{p=Prime[n], k=1}, While[!PrimeQ[p+2^k], k++]; k]; Join[{0}, Array[sk, 110, 2]] (* Harvey P. Dale, Jul 07 2013 *)

Prog

(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, return(p))); print(); print(s); } \\ Cino Hilliard, Jun 27 2005

Crossrefs

Cf. A067760, A137715.

Sequence in context: A083269 A097306 A102632 * A089611 A248470 A082067

Adjacent sequences: A094073 A094074 A094075 * A094077 A094078 A094079

Keyword

nonn

Author

Reinhard Zumkeller, Apr 29 2004

Extensions

More terms from Don Reble, May 02 2004

More terms from Cino Hilliard, Jun 27 2005

More terms from Charles R Greathouse IV, Feb 08 2008

Status

approved