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

Author

Jianing Song, May 13 2024

Keywords

• nonn

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.

Examples

			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.
		

Links

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

Crossrefs

Cf. A211244.

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

Programs

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 *)

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)))