A257379 Smallest odd number k such that k*n*2^n - 1 is a prime number.

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

Offset

1,1

Comments

Conjecture: a(n) exists for every n.

The conjecture follows from Dirichlet's theorem on primes in arithmetic progressions. - Robert Israel, Jan 05 2016

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

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

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.

Links

Pierre CAMI, Table of n, a(n) for n = 1..10000

Example

1*1*2^1 - 1 = unity, 3*1*2^1 - 1 = 5, which is prime, so a(1) = 3.
1*2*2^2 - 1 = 7, which is prime, so a(2) = 1.
1*3*2^3 - 1 = 23, which is prime, so a(3) = 1.

Maple

Q:= proc(m) local k;
  for k from 1 by 2 do if isprime(k*m-1) then return k fi od
end proc:
seq(Q(n*2^n), n=1..100); # Robert Israel, Jan 05 2016

Mathematica

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

Prog

(PFGW & SCRIPT)
SCRIPT
DIM n, 0
DIM k
DIMS t
OPENFILEOUT myf, a(n).txt
LABEL loop1
SET n, n+1
IF n>3000 THEN END
SET k, -1
LABEL loop2
SET k, k+2
SETS t, %d, %d\,; n; k
PRP k*n*2^n-1, t
IF ISPRP THEN GOTO a
GOTO loop2
LABEL a
WRITE myf, t
GOTO loop1
(PARI) a(n) = k=1; while(!isprime(k*n*2^n-1), k+=2); k \\ Colin Barker, Apr 21 2015

Crossrefs

Cf. A002234, A256787, A257378, A266909.

Sequence in context: A176149 A091442 A025834 * A336456 A227898 A035649

Adjacent sequences: A257376 A257377 A257378 * A257380 A257381 A257382

Keyword

nonn

Author

Pierre CAMI, Apr 21 2015

Status

approved