A350703 a(n) is the least integer k such that (2*n*k+1) | (2^k-1).

3, 18, 5, 9, 15, 50, 40, 16, 7, 156, 60, 25, 180, 102, 113, 81, 10, 50, 29, 159, 51, 56, 24, 36, 47, 90, 337, 72, 55, 106, 33, 102, 780, 28, 117, 25, 155, 540, 60, 104, 223, 1012, 168, 180, 91, 540, 3132, 47, 510, 412, 154, 45, 80, 432, 201, 36, 90, 144, 97, 53, 279, 880

Offset: 1

Author

Karl-Heinz Hofmann, Feb 03 2022

Keywords

• nonn

Comments

The formula 2nk+1 is used to find trivial factors of Mersenne(p). Here it is used for all exponents (prime exponents and not prime exponents).

Mersenne primes of A000043 can be found in this sequence too (except for 2). E.g.: a(1, 3, 9, 315, 3855, 13797) = A000043(2..7).

If n mod 4 = 2 then a(n) must be composite.

Examples

			a(5) = 15: 2^15 - 1 = 32767; 2*5*15 + 1 = 151; 32767 mod 151 = 0, and there are no numbers < 15 which satisfy the requirement for n = 5.
		

Links

• Karl-Heinz Hofmann, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000043, A002515, A188130, A122095, A188133, A350702.

Programs

Mathematica

a[n_] := Module[{k = 1}, While[PowerMod[2, k, 2*n*k + 1] != 1, k++]; k];  Array[a, 62] (* Amiram Eldar, Feb 03 2022 *)

PARI

a(n) = my(k=1); while (Mod(2, 2*n*k+1)^k != 1, k++); k; \\ Michel Marcus, Feb 03 2022

Python

def A350703(k,expo):
    while pow(2, expo, 2*k*expo+1) != 1: expo += 1
    return expo
print([A350703(k,1) for k in range(1, 63)])