A372800 Smallest prime p such that the multiplicative order of 16 modulo p is 2*n, or 0 if no such prime exists.

3, 5, 31, 17, 151, 109, 631, 113, 127, 1181, 331, 433, 13963, 1709, 3331, 1217, 2687, 397, 1103, 241, 2143, 1013, 18539, 1777, 2351, 4421, 2971, 673, 3191, 3061, 683, 257, 58147, 1429, 38431, 1657, 11471, 22573, 49999, 3121, 17467, 33013, 252583, 1321, 23671, 51797, 26227, 4513

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)/(2*n) in base 16. Also the first prime p such that {k/p : 1 <= k <= p-1} has 2*n different cycles when written out in base 16.

Since ord(a^m,k) = ord(a,k)/gcd(m,ord(a,k)) for gcd(a,k) = 1, we have that (p-1)/ord(16,p) = ((p-1)/ord(2,p)) * gcd(4,ord(2,p)) is always even. Here ord(a,k) is the multiplicative order of a modulo k.

Examples

			In the following examples let () denote the reptend. The prime numbers themselves and the fractions are written out in decimal.
The base-16 expansion of 1/3 is 0.(5), so the reptend has length 1 = (3-1)/2. Also, the base-16 expansions of 1/3 = 0.(5) and 2/3 = 0.(A) have two cycles 5 and A. 3 is the smallest such prime, so a(1) = 3.
The base-16 expansion of 1/5 is 0.(3), so the reptend has length 1 = (5-1)/4. Also, the base-16 expansions of 1/5 = 0.(3), 2/5 = (0.6), 3/5 = 0.(9) and 4/5 = 0.(C) have four cycles 3, 6, 9 and A. 5 is the smallest such prime, so a(2) = 5.
The base-16 expansion of 1/31 is 0.(08421), so the reptend has length 5 = (31-1)/6. Also, the base-16 expansions of 1/31, 2/31, ..., 30/31 have six cycles 08421, 18C63, 294A5, 39CE7, 5AD6B and 7BDEF. 31 is the smallest such prime, so a(3) = 31.
		

Links

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

Crossrefs

Cf. A372801.

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

Programs

Mathematica

a[n_] := a[n] = For[p = 2, True, p = NextPrime[p], If[MultiplicativeOrder[16, p] == (p-1)/(2n), Return[p]]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 24 2024 *)

PARI

a(n,{base=16}) = forprime(p=2, oo, if((base%p) && znorder(Mod(base,p)) == (p-1)/(n * if(issquare(base), 2, 1)), return(p)))