A005939 Pseudoprimes to base 10.

9, 33, 91, 99, 259, 451, 481, 561, 657, 703, 909, 1233, 1729, 2409, 2821, 2981, 3333, 3367, 4141, 4187, 4521, 5461, 6533, 6541, 6601, 7107, 7471, 7777, 8149, 8401, 8911, 10001, 11111, 11169, 11649, 12403, 12801, 13833, 13981, 14701, 14817, 14911, 15211

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This sequence is a subsequence of A121014 & A121912. In fact the terms are composite terms n of these sequences such that gcd(n,10)=1. Theorem: If both numbers q & 2q-1 are primes(q is in the sequence A005382) and n=q*(2q-1) then 10^(n-1) == 1 (mod n) (n is in the sequence A005939) iff mod(q, 20) is in the set {1, 7, 19}. 91,703,12403,38503,79003,188191,269011,... are such terms. - Farideh Firoozbakht, Sep 15 2006

Composite numbers n such that 10^(n-1) == 1 (mod n). - Michel Lagneau, Feb 18 2012

Composite numbers n such that the number of digits of the period of 1/n divides n-1. - Davide Rotondo, Dec 16 2020

References

• R. K. Guy, Unsolved Problems in Number Theory, A12.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• C. Pomerance & N. J. A. Sloane, Correspondence, 1991
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001567 (pseudoprimes to base 2), A005382, A121014, A121912.

Programs

Mathematica

Select[Range[15300], ! PrimeQ[ # ] && PowerMod[10, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)