A020138 Pseudoprimes to base 9.

4, 8, 28, 52, 91, 121, 205, 286, 364, 511, 532, 616, 671, 697, 703, 946, 949, 1036, 1105, 1288, 1387, 1541, 1729, 1891, 2465, 2501, 2665, 2701, 2806, 2821, 2926, 3052, 3281, 3367, 3751, 4376, 4636, 4961, 5356, 5551, 6364, 6601, 6643, 7081, 7381, 7913, 8401

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

This sequence is a subsequence of A122786. In fact the terms are composite terms n of A122786 such that gcd(n,3)=1. Theorem: If both numbers q & 2q-1 are primes greater than 3 and n=q*(2q-1) then 9^(n-1)==1 (mod n) (n is in the sequence). So for n>2 A005382(n)* (2*A005382(n)-1) is in the sequence; 91,703,1891,2701,12403,18721,... is the related subsequence. - Farideh Firoozbakht, Sep 15 2006

Composite numbers n such that 9^(n-1) == 1 (mod n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..159 from R. J. Mathar, terms 160..1000 from T. D. Noe)
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001567 (pseudoprimes to base 2), A005382, A122786.

Programs

Mathematica

Select[Range[8500], ! PrimeQ[ # ] && PowerMod[9, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)