A057778 a(n) is the least odd k such that k*2^n + 1 is prime.

1, 1, 1, 5, 1, 3, 3, 5, 1, 15, 13, 9, 3, 5, 7, 5, 1, 9, 3, 11, 7, 11, 25, 45, 45, 5, 7, 15, 13, 23, 3, 35, 43, 9, 75, 59, 3, 15, 15, 5, 27, 3, 9, 9, 15, 35, 19, 27, 15, 23, 7, 17, 7, 51, 49, 5, 27, 29, 99, 27, 31, 53, 105, 9, 25, 9, 3, 9, 31, 23, 39, 39, 127, 23, 67, 5, 93, 29, 15, 249

Offset: 0

Author

Labos Elemer, Nov 02 2000

Keywords

• nonn

Comments

There are no Sierpiński numbers in the sequence. See A076336. - Thomas Ordowski, Aug 13 2017

Conjecture: for n > 0, a(n) = k < 2^n, so k*2^n + 1 is a Proth prime A080076. - Thomas Ordowski, Apr 13 2019

Examples

			For n = 10, the first primes in the 1024k + 1 arithmetic progression occur at k = 12, 13, 15, 18, 19, ...; 13 is the first odd number, so a(10)=13, while A035050(10)=12. The corresponding primes are 12289 and 13313.
For n = 79, the first primes in the (2^79)k + 1 = 604462909807314587353088k + 1 progression occur at k = 36, 44, 104, 249, 296, 299, so a(79)=249, the first odd number, while A035050(79)=36. The two primes arising are 21760664753063325144711169 and 150511264542021332250918913, respectively.
		

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Poo-Sung Park, Multiplicative functions with f(p + q - n_0) = f(p) + f(q) - f(n_0), arXiv:2002.09908 [math.NT], 2020.
• Index entries for sequences related to primes in arithmetic progressions

Crossrefs

Terms are not necessarily in A035050.

Cf. A006093, A035050, A076336, A080076 A085427, A126717.

Programs

Mathematica

Table[k = 1; While[! PrimeQ[k 2^n + 1], k += 2]; k, {n, 0, 80}] (* Michael De Vlieger, Jul 04 2016 *)

PARI

a(n) = k=1; while(!isprime(k*2^n+1), k+=2); k; \\ Michel Marcus, Dec 10 2013

Formula

a(n) = Min{k: 1+2^n*k is prime and k is odd}.

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

Conjecture: a(n) = O(n*log(n)). - Thomas Ordowski, Oct 16 2014