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A007138 Smallest primitive factor of 10^n - 1. Also smallest prime p such that 1/p has repeating decimal expansion of period n.
(Formerly M2888)
3, 11, 37, 101, 41, 7, 239, 73, 333667, 9091, 21649, 9901, 53, 909091, 31, 17, 2071723, 19, 1111111111111111111, 3541, 43, 23, 11111111111111111111111, 99990001, 21401, 859, 757, 29, 3191, 211, 2791, 353, 67, 103, 71, 999999000001 (list; graph; refs; listen; history; text; internal format)



In the 18th century, the Japanese mathematician Ajima Naonobu (a.k.a. Ajima Chokuyen) gave the first 16 terms (Smith and Mikami, p. 199). - Jonathan Sondow, May 25 2013

Also the least prime number p such that the multiplicative order of 10 modulo p is n. - Robert G. Wilson v, Dec 09 2013

n always divides p-1. - Jon Perry, Nov 02 2014


Ajima Naonobu (aka Ajima Chokuyen), Fujin Isshũ (Periods of Decimal Fractions).

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

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


T. D. Noe, Table of n, a(n) for n=1..500

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

Torbjörn Granlund, Factors of 10^n-1.

Makoto Kamada, Factorizations of 11...11 (Repunit).

Yousuke Koide, Factors of Repunit Numbers.

David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; chapter X.

S. S. Wagstaff, Jr., The Cunningham Project

Eric Weisstein's World of Mathematics, Decimal Expansion

Wikipedia, Repeating decimals

Robert G. Wilson v, Smallest primitive factor of 10^n -1, or 0 if not yet found, for a(n) and n=1..10000

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


a(3) = 37 since 1/37 = 0.027027... has period 3, and 37 is the smallest such prime (in fact, the only one).


S:= {}:

for n from 1 to 60 do

  F:= numtheory:-factorset(10^n-1) minus S;

  A[n]:= min(F);

  S:= S union F;


seq(A[n], n=1..60); # Robert Israel, Nov 10 2014


Cf. A046107.

Cf. A001913.

Sequence in context: A069358 A108544 A095088 * A046107 A243110 A061075

Adjacent sequences:  A007135 A007136 A007137 * A007139 A007140 A007141




N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v, Jud McCranie



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Last modified January 17 16:11 EST 2019. Contains 319235 sequences. (Running on oeis4.)