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A285300 Numbers k such that 3^(k-1) == 2^(k-1) !== 1 (mod k).

%I A285300 #50 Feb 25 2021 01:55:23
%S A285300 65,133,529,793,1649,2059,2321,4187,5185,6305,6541,6697,6817,7471,
%T A285300 7613,8113,10963,11521,13213,13333,13427,14701,14981,19171,19201,
%U A285300 19909,21349,21667,22177,26065,26467,32873,35443,36569,37333,38897,42121,42127,44023,47081
%N A285300 Numbers k such that 3^(k-1) == 2^(k-1) !== 1 (mod k).
%C A285300 All terms are odd composite numbers. There are no pseudoprimes to bases 2 or 3 in this sequence.
%C A285300 Are there infinitely many numbers of this kind?
%C A285300 From _Max Alekseyev_, Apr 16 2017: (Start)
%C A285300 Also, Fermat pseudoprimes base 2/3 that are not Fermat pseudoprimes base 2.
%C A285300 Also, the set difference of A073631 and either of ({1} U A001567), ({1} U A005935), or ({1} U A052155). (End)
%H A285300 Amiram Eldar, <a href="/A285300/b285300.txt">Table of n, a(n) for n = 1..1000</a>
%e A285300 2^64 = 18446744073709551616 = 65 * 283796062672454640 + 16 and 3^64 = 3433683820292512484657849089281 = 65 * 52825904927577115148582293681 + 16. Therefore 65 is in the sequence.
%e A285300 Note: a(3) = 529 = 23^2 and a(40) = 47081 = 23^2 * 89.
%p A285300 filter:= proc(n) local t;
%p A285300   t:= 3 &^(n-1) mod n;
%p A285300   if t = 1 then return false fi;
%p A285300   t = 2 &^(n-1) mod n;
%p A285300 end proc:
%p A285300 select(filter, [seq(i,i=3..10^5,2)]); # _Robert Israel_, Apr 27 2017
%t A285300 Select[Range[2, 10^5], PowerMod[2, # - 1, #] == PowerMod[3, # - 1, #] != 1 &] (* _Giovanni Resta_, Apr 16 2017 *)
%o A285300 (PARI) is(n) = Mod(3, n)^(n-1)==2^(n-1) && Mod(2, n)^(n-1)!=1 \\ _Felix Fröhlich_, Apr 27 2017
%Y A285300 Cf. A001567, A005935, A052155, A073631.
%K A285300 nonn
%O A285300 1,1
%A A285300 _Thomas Ordowski_, Apr 16 2017
%E A285300 More terms from _Giovanni Resta_, Apr 16 2017