A344681 a(n) is the smallest k > n such that 2^(k-n) == 1 (mod k).

1, 3, 20737, 9, 7, 25, 31, 15, 127, 17, 73, 15, 23, 33, 3479, 21, 31, 65, 131071, 51, 524287, 31, 127, 33, 47, 69, 31, 39, 49, 43, 233, 87, 1361567, 45, 89, 51, 71, 73, 223, 57, 79, 65, 13367, 51, 431, 63, 73, 69, 2351, 97, 127, 63, 103, 65, 6361, 73, 89, 63, 721, 87, 179951

Offset

0,2

Comments

Smallest odd k > n such that 2^k == 2^n (mod k).

a(n) is the smallest odd k > n such that A002326((k-1)/2) divides k-n.

If a(n) is a prime p, then 2^(n-1) == 1 (mod p).

Note that a(2n+2) <= A002184(n) for n > 0.

If p is an odd prime, then a(p) <= p^2.

Links

Michel Marcus, Table of n, a(n) for n = 0..149

Mathematica

a[0] = 1; a[n_] := Module[{k = n + 1}, While[PowerMod[2, k - n, k] != 1, k++];
k]; Array[a, 60, 0] (* Amiram Eldar, Aug 17 2021 *)

Prog

(Python)
def a(n):
    if n == 0: return 1
    k = n + 1
    while pow(2, k-n, k) != 1: k += 1
    return k
print([a(n) for n in range(61)]) # Michael S. Branicky, Aug 17 2021
(PARI) a(n) = my(k=n+1); while(Mod(2, k)^(k-n) != 1, k++); k; \\ Michel Marcus, Aug 17 2021

Crossrefs

Cf. A002184, A002326, A346988.

Sequence in context: A203182 A128121 A112406 * A171364 A307670 A115475

Adjacent sequences: A344678 A344679 A344680 * A344682 A344683 A344684

Keyword

nonn

Author

Thomas Ordowski, Aug 17 2021

Extensions

More terms from Amiram Eldar, Aug 17 2021

Status

approved