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Smallest odd number k such that k*n*2^n - 1 is a prime number.
4

%I #31 Jan 06 2016 17:06:23

%S 3,1,1,3,3,1,3,3,5,5,9,5,7,7,3,17,11,11,7,9,11,15,3,7,9,67,3,45,3,1,

%T 33,21,15,23,17,3,7,9,19,15,17,63,51,3,9,33,53,61,13,45,75,39,83,43,7,

%U 19,13,41,5,19,31,165,13,27,3,13,135,33,31,15,33,87

%N Smallest odd number k such that k*n*2^n - 1 is a prime number.

%C Conjecture: a(n) exists for every n.

%C The conjecture follows from Dirichlet's theorem on primes in arithmetic progressions. - _Robert Israel_, Jan 05 2016

%C As N increases, (Sum_{n=1..N} k) / (Sum_{n=1..N} n) approaches 0.833.

%C If k=1 then n*2^n-1 is a Woodall prime (see A002234).

%C Generalized Woodall primes have the form n*b^n-1, I propose to name the primes k*n*2^n-1 generalized Woodall primes of the second type.

%H Pierre CAMI, <a href="/A257379/b257379.txt">Table of n, a(n) for n = 1..10000</a>

%e 1*1*2^1 - 1 = unity, 3*1*2^1 - 1 = 5, which is prime, so a(1) = 3.

%e 1*2*2^2 - 1 = 7, which is prime, so a(2) = 1.

%e 1*3*2^3 - 1 = 23, which is prime, so a(3) = 1.

%p Q:= proc(m) local k;

%p for k from 1 by 2 do if isprime(k*m-1) then return k fi od

%p end proc:

%p seq(Q(n*2^n),n=1..100); # _Robert Israel_, Jan 05 2016

%t Table[k = 1; While[!PrimeQ[k*n*2^n - 1], k += 2]; k, {n, 72}] (* _Michael De Vlieger_, Apr 21 2015 *)

%o (PFGW & SCRIPT)

%o SCRIPT

%o DIM n, 0

%o DIM k

%o DIMS t

%o OPENFILEOUT myf, a(n).txt

%o LABEL loop1

%o SET n, n+1

%o IF n>3000 THEN END

%o SET k, -1

%o LABEL loop2

%o SET k, k+2

%o SETS t, %d, %d\,; n; k

%o PRP k*n*2^n-1, t

%o IF ISPRP THEN GOTO a

%o GOTO loop2

%o LABEL a

%o WRITE myf, t

%o GOTO loop1

%o (PARI) a(n) = k=1; while(!isprime(k*n*2^n-1), k+=2); k \\ _Colin Barker_, Apr 21 2015

%Y Cf. A002234, A256787, A257378, A266909.

%K nonn

%O 1,1

%A _Pierre CAMI_, Apr 21 2015