A191620 - OEIS

%I #18 May 20 2023 12:54:14

%S 0,1,2,1,1,2,2,7,1,1,14,2,11,11,2,22,7,1,2,8,2,11,14,32,2,13,2,11,52,

%T 8,10,49,13,11,4,11,31,1,23,64,11,47,20,38,1,14,4,88,7,1,47,14,22,8,2,

%U 14,1,31,20,71,20,4,44,101,38,43,80,49,11,59,4,8

%N Least k such that (2^n-k)*2^n - 1 is a prime number

%C Does a(n) exist for every n?  This does not seem to be known, even on the GRH; see Heath-Brown. [_Charles R Greathouse IV_, Dec 27 2011]

%H Pierre CAMI, <a href="/A191620/b191620.txt">Table of n, a(n) for n = 1..5000</a>

%H D. R. Heath-Brown, <a href="https://web.archive.org/web/20160317102203/http://eprints.maths.ox.ac.uk/166/1/linnik.pdf">Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression</a>. Proceedings of the London Mathematical Society 64:3 (1992), pp. 265-338.

%t Table[a = 0; While[! PrimeQ[(2^n - a)*2^n - 1], a++]; a, {n, 100}] (* _T. D. Noe_, Jun 11 2011 *)

%o (PARI) a(n)=forstep(k=4^n-1,1,-2^n,if(ispseudoprime(k),return(2^n-(k+1)>>n))) \\ _Charles R Greathouse IV_, Dec 27 2011

%Y Cf. A191617, A191618, A191619, A191621.

%K nonn

%O 1,3

%A _Pierre CAMI_, Jun 09 2011