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
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