A153508 Sarrus numbers A001567 that are not Carmichael numbers A002997.

341, 645, 1387, 1905, 2047, 2701, 3277, 4033, 4369, 4371, 4681, 5461, 7957, 8321, 8481, 10261, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 30121, 30889, 31417, 31609, 31621, 33153, 34945

Offset: 1

Author

Artur Jasinski, Dec 28 2008

Keywords

• nonn

Comments

A composite number n is a Fermat pseudoprime to base b if and only if b^(n-1) == 1 (mod n). Fermat pseudoprimes to base 2 are sometimes called Poulet numbers, Sarrus numbers, or frequently just pseudoprimes. For any given base pseudoprimes will contain Carmichael numbers as a subset. This sequence consists of base-2 Fermat pseudoprimes without the Carmichael numbers.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..306 from Brad Clardy)

Crossrefs

Cf. A001567, A002997.

Programs

Magma

for n:= 3 to 1052503 by 2 do
  if (IsOne(2^(n-1) mod n)
      and not IsPrime(n)
      and not n mod CarmichaelLambda(n) eq 1)
      then n;
      end if;
end for; // Brad Clardy, Dec 25 2014

Maple

filter:= proc(n)
local q;
   if isprime(n) then return false fi;
   if 2 &^(n-1) mod n <> 1 then return false fi;
   if not numtheory:-issqrfree(n) then return true fi;
   for q in numtheory:-factorset(n) do
     if (n-1) mod (q-1) <> 0 then return true fi;
   od:
   false
end proc:
select(filter, [$1..10^5]); # Robert Israel, Dec 29 2014

Mathematica

Select[Range[3, 35000, 2], !PrimeQ[#] && PowerMod[2, # - 1, # ] == 1 && !Divisible[# - 1, CarmichaelLambda[#]] &] (* Amiram Eldar, Jun 25 2019 *)