A381494 Smallest number with reciprocal of period length n in base 7.

1, 2, 4, 9, 5, 2801, 36, 29, 64, 27, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 108, 419, 55, 261, 23, 47, 73, 2551, 53, 81, 145, 59, 99, 311, 256, 3631, 56036, 81229, 135, 223, 1676, 486643, 41, 83, 1017, 166003607842448777, 115, 837, 188, 13722816749522711, 153, 3529, 10204

Offset

0,2

Comments

For n > 1, a(n) is the smallest positive d such that d divides 7^n - 1 and does not divide any of 7^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

Robert Israel, Table of n, a(n) for n = 0..126

Formula

Conjecture: a(n) = A218358(n) for n>=2. - R. J. Mathar, Mar 03 2025

Example

a(3)=9 since 1/9 has period 3 in base 7 (.053053053...) and no smaller number has this property.

Maple

f:= proc(n) uses priqueue; local F, nF, pq, i, j, t, d;
F:= ifactors(7^n-1)[2]; nF:= nops(F);
initialize(pq);
for i from 1 to nF do insert([-F[i, 1], subsop(i=1, [0$nF]), NumberTheory:-MultiplicativeOrder(7, F[i, 1])], pq) od;
do
  t:= extract(pq);
  d:= -t[1];
  if t[3] = n then return d fi;
  for i from nF to 1 by -1 while t[2][i] = 0 do od:
  if t[2][i] < F[i, 2] then
     insert([t[1]*F[i, 1], subsop(i=t[2][i]+1, t[2]), ilcm(NumberTheory:-MultiplicativeOrder(7, F[i, 1]^(t[2][i]+1)), t[3])], pq)
  fi;
  for j from i+1 to nF do
    insert([t[1]*F[j, 1], subsop(j = 1, t[2]), ilcm(NumberTheory:-MultiplicativeOrder(7, F[j, 1]), t[3])], pq)
  od;
od;
FAIL
end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..60]); # Robert Israel, Mar 01 2026

Mathematica

a[n_] := First[Select[Divisors[7^n - 1], MultiplicativeOrder[7, #] == n &, 1]];
a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 50}]

Prog

(Python)
from sympy import divisors
def A381494(n): return next(d for d in divisors(7**n-1) if d>1 and all(pow(7, k, d)!=1 for k in range(1, n))) if n else 1 # Chai Wah Wu, Feb 28 2025

Crossrefs

Cf. A003060, A379640.

Sequence in context: A161360 A230242 A104654 * A011182 A392802 A304753

Adjacent sequences: A381491 A381492 A381493 * A381495 A381496 A381497

Keyword

nonn,base

Author

Erich Friedman, Feb 25 2025

Status

approved