A126717 Least odd k such that k*2^n-1 is prime.

3, 3, 1, 1, 3, 1, 3, 1, 5, 7, 5, 3, 5, 1, 5, 9, 17, 1, 3, 1, 17, 7, 33, 13, 39, 57, 11, 21, 27, 7, 213, 1, 5, 31, 3, 25, 17, 21, 3, 25, 107, 15, 33, 3, 35, 7, 23, 31, 5, 19, 11, 21, 65, 147, 5, 3, 33, 51, 77, 45, 17, 1, 53, 9, 3, 67, 63, 43, 63, 51, 27, 73, 5, 15, 21, 25, 3, 55, 47, 69

Offset

0,1

Comments

If a(n)=1 then n is a Mersenne exponent (A000043). - Pierre CAMI, Apr 22 2013

From Pierre CAMI, Apr 03 2017: (Start)

Empirically, as N increases, (Sum_{n=1..N} a(n)) / (Sum_{n=1..N} n) tends to log(2); this is consistent with the prime number theorem as the probability that x*2^n - 1 is prime is ~ 1/(n*log(2)) if n is large enough.

For n=1 to 10000, a(n)/n < 7.5.

a(n)*2^n - 1 and a(n)*2^n + 1 are twin primes for n = 1, 2, 6, 18, 22, 63, 211, 282, 546, 726, 1032, 1156, 1321, 1553, 2821, 4901, 6634, 8335, 8529; corresponding values of a(n) are 3, 1, 3, 3, 33, 9, 9, 165, 297, 213, 177, 1035, 1065, 291, 6075, 2403, 2565, 4737, 3975, 459. (End)

Links

Pierre CAMI, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)

Ray Ballinger, Proth Search Page

Poo-Sung Park, Multiplicative functions with f(p + q - n_0) = f(p) + f(q) - f(n_0), arXiv:2002.09908 [math.NT], 2020.

Formula

a(n) << 19^n by Xylouris' improvement to Linnik's theorem. - Charles R Greathouse IV, Dec 10 2013

Conjecture: a(n) = O(n log n). - Thomas Ordowski, Oct 15 2014

Example

a(10)=5 because 5*2^10-1 is prime but 1*2^10-1 and 3*2^10-1 are not.

Mathematica

f[n_] := Block[{k = 1}, While[ !PrimeQ[k*2^n - 1], k += 2]; k]; Table[f@n, {n, 0, 80}] (* Robert G. Wilson v, Feb 20 2007 *)

Prog

(PARI) a(n) = {my(k=1); while(!isprime(k*2^n - 1), k+=2); k}; \\ Indranil Ghosh, Apr 03 2017
(Python)
from sympy import isprime
def a(n):
    k=1
    while True:
        if isprime(k*2**n - 1): return k
        k+=2
print([a(n) for n in range(101)]) # Indranil Ghosh, Apr 03 2017

Crossrefs

Cf. A035050, A057778, A085427, A284631.

Sequence in context: A371741 A303992 A365213 * A124039 A379488 A350617

Adjacent sequences: A126714 A126715 A126716 * A126718 A126719 A126720

Keyword

nonn

Author

Bernardo Boncompagni, Feb 13 2007

Extensions

More terms from Robert G. Wilson v, Feb 20 2007

Status

approved