

A126717


Least odd k such that k*2^n1 is prime.


11



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
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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


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*2^101 is prime but 1*2^101 and 3*2^101 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 xrange(101)] # Indranil Ghosh, Apr 03 2017


CROSSREFS

Cf. A035050, A057778, A085427, A284631.
Sequence in context: A214281 A125300 A303992 * A124039 A096433 A084101
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



