A122913 Minimum numbers k such that (k^2*2^n + 1) is prime.

1, 1, 3, 1, 6, 2, 3, 1, 6, 5, 3, 4, 12, 2, 6, 1, 3, 10, 15, 5, 9, 5, 18, 25, 9, 13, 9, 14, 12, 7, 6, 9, 3, 17, 9, 9, 15, 12, 9, 6, 6, 3, 3, 11, 42, 18, 21, 9, 66, 10, 33, 5, 27, 7, 48, 80, 24, 40, 12, 20, 6, 10, 3, 5, 3, 7, 3, 79, 75, 63, 96, 40, 48, 20, 24, 10, 12, 5, 6, 15, 3, 22, 72, 11

Offset: 1

Author

Alexander Adamchuk, Sep 18 2006

Keywords

• nonn

Comments

3 divides a(2k+1) for k>0. Corresponding primes of the form (k^2*2^n + 1) are listed in A122912[n] = {3,5,73,17,1153,257,1153,257,18433,25601,18433,65537,1179649,65537,1179649,65537,1179649,26214401,117964801,...}. There are repeating patterns in a(n) such that for many n a(n) = 2*a(n+2) and a(n+1) = 2*a(n+3). For example, {6,2,3,1}, {12,2,6,1}, {42,18,21,9}, {96,40,48,20,24,10,12,5,6}, {66,10,33,5}, {48,80,24,40,12,20,6,10,3}, {366,38,183,19}. These patterns correspond to identical twin runs in A122912[n] such that A122912[n] = A122912[n+2] and A122912[n+1] = A122912[n+3]. The final index of many such twin runs is perfect power such as {8,16,25,64,81,100,...}.

Crossrefs

Cf. A112912.

Programs

Mathematica

mnk[n_]:=Module[{k=1,c=2^n},While[!PrimeQ[k^2 c+1],k++];k]; Array[mnk,90] (* Harvey P. Dale, Jul 22 2025 *)

Formula

a(n) = Sqrt[ (A122912[n] - 1) / 2^n ].