A176494 Least m >= 1 for which |2^m - prime(n)| is prime.

3, 1, 1, 2, 1, 2, 1, 2, 4, 1, 3, 2, 1, 2, 4, 4, 1, 3, 2, 1, 3, 2, 4, 3, 2, 1, 2, 1, 2, 47, 2, 6, 1, 8, 1, 3, 5, 2, 4, 4, 1, 6, 1, 2, 1, 5, 5, 2, 1, 2, 4, 1, 8, 4, 6, 8, 1, 3, 2, 1, 4, 7, 2, 1, 2, 9, 791, 4, 1, 2, 8, 3, 9, 5, 2, 4, 3, 2, 3, 8, 1, 6, 1, 3, 2, 4, 3, 2, 1, 2, 4, 3, 2, 3

Offset: 2

Author

Vladimir Shevelev, Apr 19 2010, Aug 15 2010

Keywords

• nonn

Comments

a(n)=1 iff p_n is second of twin primes (A006512); for n > 4, a(n)=2 iff p_n is second of cousin primes (A046132). It is interesting to continue this sequence in order to find big jumps such as a(31)-a(30). Is it true that such jumps can be arbitrarily large, either (a) in the sense of differences a(n+1)-a(n), or (b) in the sense of ratios a(n+1)/a(n)?

Conjecture. For every odd prime p, the sequence {|2^n - p|} contains at least one prime. The record values of the sequence appear at n = 2, 10, 31, 68, 341, ... and are 3, 4, 47, 791, ... Note that up to now the value a(341) is not known. Charles R Greathouse IV calculated the following two values: a(815)=16464, a(591)=58091 and noted that a(341) is much larger [private communication, May 27 2010]. - Vladimir Shevelev, May 29 2010

Links

• Amiram Eldar, Table of n, a(n) for n = 2..340

Crossrefs

Cf. A176303, A175347, A006512, A046132.

Programs

Mathematica

lm[n_]:=Module[{m=1},While[!PrimeQ[Abs[2^m-n]],m++];m]; Table[lm[i],{i,Prime[ Range[2,100]]}] (* Harvey P. Dale, Aug 11 2014 *)

Extensions

Beginning with a(31), the terms were calculated by Zak Seidov - private communication, Apr 20 2010

Sequence extended by R. J. Mathar via the Seqfan Discussion List, Aug 15 2010