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

1,1

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)

Links

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A001567, A005935, A052155, A073631.

Sequence in context: A357651 A294169 A073631 * A194002 A092226 A211255

Adjacent sequences: A285297 A285298 A285299 * A285301 A285302 A285303

Keyword

nonn

Author

Thomas Ordowski, Apr 16 2017

Extensions

More terms from Giovanni Resta, Apr 16 2017

Status

approved