A007138 Smallest primitive factor of 10^n - 1. Also smallest prime p such that 1/p has repeating decimal expansion of period n.

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, 2028119, 909090909090909091

Offset: 1

Author

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

Keywords

• nonn
• base

Comments

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

Examples

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

References

• 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).

Links

• Max Alekseyev, Table of n, a(n) for n = 1..438 (terms 1..364 from T. D. Noe)
• 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 and Max Alekseyev, 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

Crossrefs

First column of A046107.

Cf. A001913.

Cf. A112927 (base 2), A143663 (base 3), A112092 (base 4), A143665 (base 5), A379639 (base 6), A379640 (base 7), A379641 (base 8), A379642 (base 9), A007138 (base 10), A379644 (base 11), A252170 (base 12).

Programs

Maple

S:= {}:
for n from 1 to 60 do
  F:= numtheory:-factorset(10^n-1) minus S;
  A[n]:= min(F);
  S:= S union F;
od:
seq(A[n],n=1..60); # Robert Israel, Nov 10 2014

Mathematica

s={}; Reap[Scan[(x=Complement[FactorInteger[10^#-1][[All,1]],s]; Sow[Min[x]]; s=Union[s,x])&,Range@60]][[2,1]] (* Shenghui Yang, Apr 15 2025 *)

Extensions

b-file truncated to 364 terms as a(365) was wrong and is currently unknown (pointed by Eric Chen), and a-file revised by Max Alekseyev, Apr 26 2022