A112505 Number of primitive prime factors of 10^n-1.

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 3, 1, 2, 2, 2, 2, 1, 2, 3, 3, 1, 1, 3, 2, 2, 3, 5, 3, 3, 5, 2, 3, 3, 1, 3, 1, 1, 2, 4, 3, 4, 3, 2, 4, 2, 1, 2, 3, 4, 2, 4, 2, 4, 2, 3, 2, 2, 3, 7, 1, 5, 4, 2, 2, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 4, 4, 6, 2, 5, 2, 3, 2, 3, 3, 3, 2, 5, 3, 7, 3, 1, 3, 5, 4, 3, 2, 4, 4

Offset: 1

Author

T. D. Noe, Sep 08 2005

Keywords

• hard
• nonn

Comments

Also the number of primes whose reciprocal is a repeating decimal of length n. The number of numbers in each row of table A046107.

By Zsigmondy's theorem, a(n) >= 1. When a(n)=1, the corresponding prime is called a unique prime (see A007498, A040017 and A051627).

Links

• Table of n, a(n) for n = 1..352
• Eric Weisstein's World of Mathematics, Primitive Prime Factor
• Eric Weisstein's World of Mathematics, Zsigmondy Theorem
• Eric Weisstein's World of Mathematics, Unique Prime

Crossrefs

Cf. A007138 (smallest primitive prime factor of 10^n-1), A102347 (number of distinct prime factors of 10^n-1), A046107.

Programs

Mathematica

pp={}; Table[f=Transpose[FactorInteger[10^n-1]][[1]]; p=Complement[f, pp]; pp=Union[pp, p]; Length[p], {n, 66}]

Extensions

Terms to a(276) in b-file from T. D. Noe, Jun 01 2010

a(277)-a(322) in b-file from Ray Chandler, May 01 2017

a(323)-a(352) in b-file from Max Alekseyev, Apr 28 2022