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

1,1

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.

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A082057 A161687 A245498 * A120647 A131635 A324554

Adjacent sequences: A350700 A350701 A350702 * A350704 A350705 A350706

Keyword

nonn

Author

Karl-Heinz Hofmann, Feb 03 2022

Status

approved