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Pseudoprimes to base 8.
7

%I #37 Jun 18 2021 17:03:43

%S 9,21,45,63,65,105,117,133,153,231,273,341,481,511,561,585,645,651,

%T 861,949,1001,1105,1281,1365,1387,1417,1541,1649,1661,1729,1785,1905,

%U 2047,2169,2465,2501,2701,2821,3145,3171,3201,3277,3605,3641,4005,4033,4097

%N Pseudoprimes to base 8.

%C 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

%C Composite numbers k such that 8^(k-1) == 1 (mod k). - _Michel Lagneau_, Feb 18 2012

%C If q and 3q-2 are odd primes, then q*(3q-2) is in the sequence. - _Davide Rotondo_, May 25 2021

%H Amiram Eldar, <a href="/A020137/b020137.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..613 from R. J. Mathar, terms 614..1000 from T. D. Noe)

%H <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>

%t Select[Range[4100], ! PrimeQ[ # ] && PowerMod[8, (# - 1), # ] == 1 &] (* _Farideh Firoozbakht_, Sep 15 2006 *)

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

%K nonn

%O 1,1

%A _David W. Wilson_