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%I #53 May 26 2020 15:26:33
%S 1,1,1,5,1,3,3,5,1,15,13,9,3,5,7,5,1,9,3,11,7,11,25,45,45,5,7,15,13,
%T 23,3,35,43,9,75,59,3,15,15,5,27,3,9,9,15,35,19,27,15,23,7,17,7,51,49,
%U 5,27,29,99,27,31,53,105,9,25,9,3,9,31,23,39,39,127,23,67,5,93,29,15,249
%N a(n) is the least odd k such that k*2^n + 1 is prime.
%C There are no Sierpiński numbers in the sequence. See A076336. - _Thomas Ordowski_, Aug 13 2017
%C Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - _Thomas Ordowski_, Apr 13 2019
%H T. D. Noe, <a href="/A057778/b057778.txt">Table of n, a(n) for n = 0..1000</a>
%H 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.
%H <a href="/index/Pri#primes_AP">Index entries for sequences related to primes in arithmetic progressions</a>
%F a(n) = Min{k: 1+2^n*k is prime and k is odd}.
%F a(n) << 19^n by Xylouris's improvement to Linnik's theorem. - _Charles R Greathouse IV_, Dec 10 2013
%F Conjecture: a(n) = O(n*log(n)). - _Thomas Ordowski_, Oct 16 2014
%e For n = 10, the first primes in the 1024k + 1 arithmetic progression occur at k = 12, 13, 15, 18, 19, ...; 13 is the first odd number, so a(10)=13, while A035050(10)=12. The corresponding primes are 12289 and 13313.
%e For n = 79, the first primes in the (2^79)k + 1 = 604462909807314587353088k + 1 progression occur at k = 36, 44, 104, 249, 296, 299, so a(79)=249, the first odd number, while A035050(79)=36. The two primes arising are 21760664753063325144711169 and 150511264542021332250918913, respectively.
%t Table[k = 1; While[! PrimeQ[k 2^n + 1], k += 2]; k, {n, 0, 80}] (* _Michael De Vlieger_, Jul 04 2016 *)
%o (PARI) a(n) = k=1; while(!isprime(k*2^n+1), k+=2); k; \\ _Michel Marcus_, Dec 10 2013
%Y Terms are not necessarily in A035050.
%Y Cf. A006093, A035050, A076336, A080076 A085427, A126717.
%K nonn
%O 0,4
%A _Labos Elemer_, Nov 02 2000
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