A046107 Triangle of prime numbers in which n-th row lists all primes p such that 1/p has decimal period n, n >= 1.

3, 11, 37, 101, 41, 271, 7, 13, 239, 4649, 73, 137, 333667, 9091, 21649, 513239, 9901, 53, 79, 265371653, 909091, 31, 2906161, 17, 5882353, 2071723, 5363222357, 19, 52579, 1111111111111111111, 3541, 27961, 43, 1933, 10838689, 23, 4093, 8779, 11111111111111111111111

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• tabf
• base

Comments

The number of numbers in each row n is given by A112505(n).

In the 18th century, the Japanese mathematician Ajima Naonobu (aka Ajima Chokuyen) gave the terms through 5882353 (Smith and Mikami, p. 199). - Jonathan Sondow, May 25 2013

Examples

			First rows of irregular triangle are:
       3;
      11;
      37;
     101;
      41,     271;
       7,      13;
     239,    4649;
      73,     137;
  333667;
    9091;
   21649,  513239;
    9901;
      53,      79, 265371653;
  909091;
      31, 2906161;
      17, 5882353;
  ...
		

References

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

Links

• Rows n=1..352, flattened
• David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; chapter X.
• Eric Weisstein's World of Mathematics, Decimal Expansion.
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A007138 (first column), A061075 (last row elements).

Programs

Mathematica

pp={}; Do[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Print[p], {n, 66}] (* T. D. Noe, Sep 08 2005 *)

Extensions

First 276 rows in b-file from T. D. Noe, Jun 01 2010

Rows n=277..322 in b-file from Ray Chandler, May 01 2017

Rows n=323..352 in b-file from Max Alekseyev, Apr 26 2022