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A126717 Least odd k such that k*2^n-1 is prime.

%I A126717 #57 Mar 22 2021 03:42:00
%S A126717 3,3,1,1,3,1,3,1,5,7,5,3,5,1,5,9,17,1,3,1,17,7,33,13,39,57,11,21,27,7,
%T A126717 213,1,5,31,3,25,17,21,3,25,107,15,33,3,35,7,23,31,5,19,11,21,65,147,
%U A126717 5,3,33,51,77,45,17,1,53,9,3,67,63,43,63,51,27,73,5,15,21,25,3,55,47,69
%N A126717 Least odd k such that k*2^n-1 is prime.
%C A126717 If a(n)=1 then n is a Mersenne exponent (A000043). - _Pierre CAMI_, Apr 22 2013
%C A126717 From _Pierre CAMI_, Apr 03 2017: (Start)
%C A126717 Empirically, as N increases, (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} n) tends to log(2); this is consistent with the prime number theorem as the probability that x*2^n - 1 is prime is ~ 1/(n*log(2)) if n is large enough.
%C A126717 For n=1 to 10000, a(n)/n < 7.5.
%C A126717 a(n)*2^n - 1 and a(n)*2^n + 1 are twin primes for n = 1, 2, 6, 18, 22, 63, 211, 282, 546, 726, 1032, 1156, 1321, 1553, 2821, 4901, 6634, 8335, 8529; corresponding values of a(n) are 3, 1, 3, 3, 33, 9, 9, 165, 297, 213, 177, 1035, 1065, 291, 6075, 2403, 2565, 4737, 3975, 459. (End)
%H A126717 Pierre CAMI, <a href="/A126717/b126717.txt">Table of n, a(n) for n = 0..10000</a>  (first 1000 terms from T. D. Noe)
%H A126717 Ray Ballinger, <a href="http://www.prothsearch.com/index.html">Proth Search Page</a>
%H A126717 Poo-Sung Park, <a href="https://arxiv.org/abs/2002.09908">Multiplicative functions with f(p + q - n_0) = f(p) + f(q) - f(n_0)</a>, arXiv:2002.09908 [math.NT], 2020.
%F A126717 a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - _Charles R Greathouse IV_, Dec 10 2013
%F A126717 Conjecture: a(n) = O(n log n). - _Thomas Ordowski_, Oct 15 2014
%e A126717 a(10)=5 because 5*2^10-1 is prime but 1*2^10-1 and 3*2^10-1 are not.
%t A126717 f[n_] := Block[{k = 1}, While[ !PrimeQ[k*2^n - 1], k += 2]; k]; Table[f@n, {n, 0, 80}] (* _Robert G. Wilson v_, Feb 20 2007 *)
%o A126717 (PARI) a(n) = {my(k=1); while(!isprime(k*2^n - 1), k+=2); k}; \\ _Indranil Ghosh_, Apr 03 2017
%o A126717 (Python)
%o A126717 from sympy import isprime
%o A126717 def a(n):
%o A126717     k=1
%o A126717     while True:
%o A126717         if isprime(k*2**n - 1): return k
%o A126717         k+=2
%o A126717 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Apr 03 2017
%Y A126717 Cf. A035050, A057778, A085427, A284631.
%K A126717 nonn
%O A126717 0,1
%A A126717 _Bernardo Boncompagni_, Feb 13 2007
%E A126717 More terms from _Robert G. Wilson v_, Feb 20 2007