A003060 Smallest number with reciprocal of period length n in decimal (base 10). (Formerly M2886)

1, 3, 11, 27, 101, 41, 7, 239, 73, 81, 451, 21649, 707, 53, 2629, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 511, 21401, 583, 243, 29, 3191, 211, 2791, 353, 67, 103, 71, 1919, 2028119, 909090909090909091

Offset

0,2

Comments

For n > 0, a(n) is the least divisor d > 1 of 10^n - 1 such that the multiplicative order of 10 mod d is n. For prime n > 3, a(n) = A007138(n). - T. D. Noe, Aug 07 2007

For n > 1, a(n) is the smallest positive d such that d divides 10^n - 1 and does not divide any of 10^k - 1 for 0 < k < n. - Maciej Ireneusz Wilczynski, Sep 06 2012, corrected by M. F. Hasler, Jun 28 2022. (For n = 1, d = 1 divides 10^n - 1 and does not divide any 10^k - 1 with 0 < k < n, but a(1) = 3 > 1.)

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.

"Cycle lengths of reciprocals", Popular Computing (Calabasas, CA), Vol. 1 (No. 4, Jul 1973), pp. 12-14.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..438

Index entries for sequences related to decimal expansion of 1/n

Mathematica

a[n_] := First[ Select[ Divisors[10^n - 1], MultiplicativeOrder[10, #] == n &, 1]]; a[0] = 1; a[1] = 3; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 13 2012, after T. D. Noe *)

Prog

(PARI) apply( {A003060(n)=!fordiv(10^n-!!n, d, d>1 && znorder(Mod(10, d))==n && return(d))}, [0..50]) \\ M. F. Hasler, Jun 28 2022

Crossrefs

Smallest primitive divisors of b^n-1: A212953 (b=2), A218356 (b=3), A218357 (b=5), A218358 (b=7), this sequence (b=10), A218359 (b=11), A218360 (b=13), A218361 (b=17), A218362 (b=19), A218363 (b=23), A218364 (b=29).

Cf. A007138, A057951, A005422, A176973.

Sequence in context: A077776 A113836 A036571 * A136983 A210611 A119074

Adjacent sequences: A003057 A003058 A003059 * A003061 A003062 A003063

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Comment corrected by T. D. Noe, Apr 15 2010

More terms from T. D. Noe, Apr 15 2010

b-file truncated at uncertain term a(439) by Max Alekseyev, Apr 30 2022

Status

approved