OFFSET
0,4
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
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.
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
EXAMPLE
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.
MATHEMATICA
Table[k = 1; While[! PrimeQ[k 2^n + 1], k += 2]; k, {n, 0, 80}] (* Michael De Vlieger, Jul 04 2016 *)
PROG
(PARI) a(n) = k=1; while(!isprime(k*2^n+1), k+=2); k; \\ Michel Marcus, Dec 10 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Nov 02 2000
STATUS
approved