A007138 - OEIS

%I M2888 #87 Nov 25 2024 19:15:19

%S 3,11,37,101,41,7,239,73,333667,9091,21649,9901,53,909091,31,17,

%T 2071723,19,1111111111111111111,3541,43,23,11111111111111111111111,

%U 99990001,21401,859,757,29,3191,211,2791,353,67,103,71,999999000001,2028119,909090909090909091

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

%C 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

%C Also the least prime number p such that the multiplicative order of 10 modulo p is n. - _Robert G. Wilson v_, Dec 09 2013

%C n always divides p-1. - _Jon Perry_, Nov 02 2014

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

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

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

%H Max Alekseyev, <a href="/A007138/b007138.txt">Table of n, a(n) for n = 1..438</a> (terms 1..364 from T. D. Noe)

%H J. Brillhart et al., <a href="http://dx.doi.org/10.1090/conm/022">Factorizations of b^n +- 1</a>, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.

%H Torbjörn Granlund, <a href="https://gmplib.org/~tege/fac10m.txt">Factors of 10^n-1</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit">Factorizations of 11...11 (Repunit)</a>.

%H Yousuke Koide, <a href="https://repunit-koide.jimdofree.com/">Factors of Repunit Numbers</a>.

%H David Eugene Smith and Yoshio Mikami, <a href="http://archive.org/details/historyofjapanes00smitiala">A History of Japanese Mathematics</a>, Open Court, Chicago, 1914; chapter X.

%H S. S. Wagstaff, Jr., <a href="http://www.cerias.purdue.edu/homes/ssw/cun/index.html">The Cunningham Project</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DecimalExpansion.html">Decimal Expansion</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Repeating_decimal">Repeating decimals</a>

%H Robert G. Wilson v and Max Alekseyev, <a href="/A007138/a007138_1.txt">Smallest primitive factor of 10^n -1, or 0 if not yet found, for a(n) and n=1..10000</a>

%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>

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

%p S:= {}:

%p for n from 1 to 60 do

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

%p   A[n]:= min(F);

%p   S:= S union F;

%p od:

%p seq(A[n],n=1..60); # _Robert Israel_, Nov 10 2014

%Y First column of A046107.

%Y Cf. A001913.

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_, _Jud McCranie_

%E 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