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%I #21 Sep 24 2019 07:10:10
%S 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,
%T 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,
%U 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
%N Least m >= 1 for which |2^m - prime(n)| is prime.
%C 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)?
%C 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
%H Amiram Eldar, <a href="/A176494/b176494.txt">Table of n, a(n) for n = 2..340</a>
%t 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 *)
%Y Cf. A176303, A175347, A006512, A046132.
%K nonn
%O 2,1
%A _Vladimir Shevelev_, Apr 19 2010, Aug 15 2010
%E Beginning with a(31), the terms were calculated by _Zak Seidov_ - private communication, Apr 20 2010
%E Sequence extended by _R. J. Mathar_ via the Seqfan Discussion List, Aug 15 2010
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