A381493 Smallest number with reciprocal of period length n in base 8.

1, 7, 3, 73, 5, 31, 19, 49, 17, 262657, 11, 23, 37, 79, 43, 631, 97, 103, 81, 32377, 25, 3577, 67, 47, 323, 601, 237, 2593, 29, 233, 209, 2147483647, 193, 199, 307, 71, 405, 223, 571, 937, 187, 13367, 817, 431, 115, 271, 139, 2351, 577, 343, 251

Offset: 0

Author

Erich Friedman, Feb 25 2025

Keywords

• nonn
• base

Comments

For n > 1, a(n) is the smallest positive d such that d divides 8^n - 1 and does not divide any of 8^k - 1 for 0 < k < n.

Examples

			a(3)=73 because 1/73 has period 3 in base 8 (.007007007...) and no smaller number has this property.
		

References

• J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Crossrefs

Cf. A003060, A379641.

Programs

Mathematica

a[n_] := First[Select[Divisors[8^n - 1], MultiplicativeOrder[8, #] == n &, 1]];
a[0] = 1; a[1] = 7; Table[a[n], {n, 0, 50}]

Python

from sympy import divisors
def A381493(n):
    if n == 0: return 1
    for d in divisors(8**n-1):
        if d>1 and all(pow(8,k,d)!=1 for k in range(1,n)):
            return d # Chai Wah Wu, Feb 28 2025