A372798 Smallest prime p such that the multiplicative order of 8 modulo p is n, or 0 if no such prime exists.

3, 17, 13, 113, 251, 7, 1163, 89, 109, 431, 1013, 577, 4421, 953, 571, 257, 4523, 127, 15467, 3761, 3109, 7151, 18539, 73, 25301, 14327, 2971, 42953, 72269, 151, 683, 12641, 331, 2687, 42701, 5113, 18797, 1103, 8581, 13121, 172283, 631, 221021, 120737, 3061, 5153, 217517

Offset

1,1

Comments

First prime p such that the expansion of 1/p has period (p-1)/n in base 8. Also the first prime p such that {k/p : 1 <= k <= p-1} has n different cycles when written out in base 8.

Links

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Example

In the following examples let () denote the reptend. The prime numbers themselves and the fractions are written out in decimal.
The base-8 expansion of 1/3 is 0.(25), so the reptend has length 2 = (3-1)/1. Also, the base-8 expansions of 1/3 = 0.(25) and 2/3 = 0.(52) have only one cycle 25. 3 is the smallest such prime, so a(1) = 3.
The base-8 expansion of 1/17 is 0.(03607417), so the reptend has length 8 = (17-1)/2. Also, the base-8 expansions of 1/17, 2/17, ..., 16/17 have two cycles 03607417 and 13226455. 17 is the smallest such prime, so a(2) = 17.
The base-8 expansion of 1/13 is 0.(0473), so the reptend has length 4 = (13-1)/3. Also, the base-8 expansions of 1/13, 2/13, ..., 12/13 have three cycles 0473, 1166 and 2354. 13 is the smallest such prime, so a(3) = 13.

Mathematica

a[n_] := a[n] = For[p = 2, True, p = NextPrime[p], If[MultiplicativeOrder[8, p] == (p-1)/n, Return[p]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 24 2024 *)

Prog

(PARI) a(n, {base=8}) = forprime(p=2, oo, if((base%p) && znorder(Mod(base, p)) == (p-1)/(n * if(issquare(base), 2, 1)), return(p)))

Crossrefs

Cf. A211244.

In other bases: A101208, A101209, A372797, A372798, A372799, A054471, A372800.

Sequence in context: A273702 A273710 A087964 * A174182 A120448 A095422

Adjacent sequences: A372795 A372796 A372797 * A372799 A372800 A372801

Keyword

nonn

Author

Jianing Song, May 13 2024

Status

approved