A005939 Pseudoprimes to base 10. (Formerly M4612)

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

1,1

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. A number is pseudoprime to base 10 if the number of digits of the period of ((n-1)!+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

Mathematica

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

Crossrefs

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

Sequence in context: A146171 A146188 A020228 * A020326 A201024 A228170

Adjacent sequences: A005936 A005937 A005938 * A005940 A005941 A005942

Keyword

nonn

Author

N. J. A. Sloane

Status

approved