A005937 Pseudoprimes to base 6.

35, 185, 217, 301, 481, 1105, 1111, 1261, 1333, 1729, 2465, 2701, 2821, 3421, 3565, 3589, 3913, 4123, 4495, 5713, 6533, 6601, 8029, 8365, 8911, 9331, 9881, 10585, 10621, 11041, 11137, 12209, 14315, 14701, 15841, 16589, 17329, 18361, 18721, 20017, 21049, 22049

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Theorem: If both numbers q and 2q-1 are primes and n=q*(2q-1) then 6^(n-1) == 1 (mod n) (n is in the sequence) iff q is of the form 12k+1. 2701, 18721, 49141, 104653, 226801, 665281, ... are such terms. This sequence is a subsequence of A122783. - Farideh Firoozbakht, Sep 12 2006

Composite numbers k such that 6^(k-1) == 1 (mod k). - Michel Lagneau, Feb 18 2012

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

• R. J. Mathar, T. D. Noe, and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (Mathar 1..118, Noe 119..1000, Greathouse 1001..10000)
• C. Pomerance & N. J. A. Sloane, Correspondence, 1991
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001567 (pseudoprimes to base 2), A122783.

Programs

Mathematica

Select[Range[20000], ! PrimeQ[ # ] && PowerMod[6, #-1, # ] == 1 &] (* Farideh Firoozbakht, Sep 12 2006 *)

Extensions

More terms from Farideh Firoozbakht, Sep 12 2006