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%I A257379 #31 Jan 06 2016 17:06:23
%S A257379 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 A257379 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 A257379 19,13,41,5,19,31,165,13,27,3,13,135,33,31,15,33,87
%N A257379 Smallest odd number k such that k*n*2^n - 1 is a prime number.
%C A257379 Conjecture: a(n) exists for every n.
%C A257379 The conjecture follows from Dirichlet's theorem on primes in arithmetic progressions. - _Robert Israel_, Jan 05 2016
%C A257379 As N increases, (Sum_{n=1..N} k) / (Sum_{n=1..N} n) approaches 0.833.
%C A257379 If k=1 then n*2^n-1 is a Woodall prime (see A002234).
%C A257379 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 A257379 Pierre CAMI, <a href="/A257379/b257379.txt">Table of n, a(n) for n = 1..10000</a>
%e A257379 1*1*2^1 - 1 = unity, 3*1*2^1 - 1 = 5, which is prime, so a(1) = 3.
%e A257379 1*2*2^2 - 1 = 7, which is prime, so a(2) = 1.
%e A257379 1*3*2^3 - 1 = 23, which is prime, so a(3) = 1.
%p A257379 Q:= proc(m) local k;
%p A257379   for k from 1 by 2 do if isprime(k*m-1) then return k fi od
%p A257379 end proc:
%p A257379 seq(Q(n*2^n),n=1..100); # _Robert Israel_, Jan 05 2016
%t A257379 Table[k = 1; While[!PrimeQ[k*n*2^n - 1], k += 2]; k, {n, 72}] (* _Michael De Vlieger_, Apr 21 2015 *)
%o A257379 (PFGW & SCRIPT)
%o A257379 SCRIPT
%o A257379 DIM n, 0
%o A257379 DIM k
%o A257379 DIMS t
%o A257379 OPENFILEOUT myf, a(n).txt
%o A257379 LABEL loop1
%o A257379 SET n, n+1
%o A257379 IF n>3000 THEN END
%o A257379 SET k, -1
%o A257379 LABEL loop2
%o A257379 SET k, k+2
%o A257379 SETS t, %d, %d\,; n; k
%o A257379 PRP k*n*2^n-1, t
%o A257379 IF ISPRP THEN GOTO a
%o A257379 GOTO loop2
%o A257379 LABEL a
%o A257379 WRITE myf, t
%o A257379 GOTO loop1
%o A257379 (PARI) a(n) = k=1; while(!isprime(k*n*2^n-1), k+=2); k \\ _Colin Barker_, Apr 21 2015
%Y A257379 Cf. A002234, A256787, A257378, A266909.
%K A257379 nonn
%O A257379 1,1
%A A257379 _Pierre CAMI_, Apr 21 2015

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