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

Author

Reinhard Zumkeller, Apr 29 2004

Keywords

• nonn

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

Examples

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

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]

Crossrefs

Cf. A067760, A137715.

Programs

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 *)

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

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