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

0,2

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.

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.

Links

Table of n, a(n) for n=0..50.

Example

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

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}]

Prog

(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

Crossrefs

Cf. A003060, A379641.

Sequence in context: A176328 A248279 A372617 * A379641 A183421 A173443

Adjacent sequences: A381490 A381491 A381492 * A381494 A381495 A381496

Keyword

nonn,base

Author

Erich Friedman, Feb 25 2025

Status

approved