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

65, 133, 529, 793, 1649, 2059, 2321, 4187, 5185, 6305, 6541, 6697, 6817, 7471, 7613, 8113, 10963, 11521, 13213, 13333, 13427, 14701, 14981, 19171, 19201, 19909, 21349, 21667, 22177, 26065, 26467, 32873, 35443, 36569, 37333, 38897, 42121, 42127, 44023, 47081

Offset: 1

Author

Thomas Ordowski, Apr 16 2017

Keywords

• nonn

Comments

All terms are odd composite numbers. There are no pseudoprimes to bases 2 or 3 in this sequence.

Are there infinitely many numbers of this kind?

From Max Alekseyev, Apr 16 2017: (Start)

Also, Fermat pseudoprimes base 2/3 that are not Fermat pseudoprimes base 2.

Also, the set difference of A073631 and either of ({1} U A001567), ({1} U A005935), or ({1} U A052155). (End)

Examples

			2^64 = 18446744073709551616 = 65 * 283796062672454640 + 16 and 3^64 = 3433683820292512484657849089281 = 65 * 52825904927577115148582293681 + 16. Therefore 65 is in the sequence.
Note: a(3) = 529 = 23^2 and a(40) = 47081 = 23^2 * 89.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A001567, A005935, A052155, A073631.

Programs

Maple

filter:= proc(n) local t;
  t:= 3 &^(n-1) mod n;
  if t = 1 then return false fi;
  t = 2 &^(n-1) mod n;
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Apr 27 2017

Mathematica

Select[Range[2, 10^5], PowerMod[2, # - 1, #] == PowerMod[3, # - 1, #] != 1 &] (* Giovanni Resta, Apr 16 2017 *)

PARI

is(n) = Mod(3, n)^(n-1)==2^(n-1) && Mod(2, n)^(n-1)!=1 \\ Felix Fröhlich, Apr 27 2017

Extensions

More terms from Giovanni Resta, Apr 16 2017