A020137 Pseudoprimes to base 8.

9, 21, 45, 63, 65, 105, 117, 133, 153, 231, 273, 341, 481, 511, 561, 585, 645, 651, 861, 949, 1001, 1105, 1281, 1365, 1387, 1417, 1541, 1649, 1661, 1729, 1785, 1905, 2047, 2169, 2465, 2501, 2701, 2821, 3145, 3171, 3201, 3277, 3605, 3641, 4005, 4033, 4097

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

This sequence is a subsequence of the sequence A122785. In fact the terms are odd composite terms of A122785. Theorem: If both numbers q and 2q-1 are primes (q is in the sequence A005382) and n=q*(2q-1) then 8^(n-1)==1 (mod n) (n is in the sequence) iff q is of the form 12k+1. 2701,18721,49141,104653,226801,665281,721801,... is the related subsequence. This subsequence is also a subsequence of the sequence A122785. - Farideh Firoozbakht, Sep 15 2006

Composite numbers k such that 8^(k-1) == 1 (mod k). - Michel Lagneau, Feb 18 2012

If q and 3q-2 are odd primes, then q*(3q-2) is in the sequence. - Davide Rotondo, May 25 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..613 from R. J. Mathar, terms 614..1000 from T. D. Noe)
• Index entries for sequences related to pseudoprimes

Crossrefs

Cf. A001567 (pseudoprimes to base 2), A005382, A122783, A122785.

Programs

Mathematica

Select[Range[4100], ! PrimeQ[ # ] && PowerMod[8, (# - 1), # ] == 1 &] (* Farideh Firoozbakht, Sep 15 2006 *)