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

Author

Bernardo Boncompagni, Feb 13 2007

Keywords

• nonn

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)

Examples

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

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.

Crossrefs

Cf. A035050, A057778, A085427, A284631.

Programs

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 *)

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

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

Extensions

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