A291601 Composite integers k such that 2^d == 2^(k/d) (mod k) for all d|k.

341, 1105, 1387, 2047, 2701, 3277, 4033, 4369, 4681, 5461, 7957, 8321, 10261, 13747, 13981, 14491, 15709, 18721, 19951, 23377, 31417, 31609, 31621, 35333, 42799, 49141, 49981, 60701, 60787, 65077, 65281, 68101, 80581, 83333, 85489, 88357, 90751, 104653, 123251, 129889

Offset

1,1

Comments

Such k must be odd.

For d=1, we have 2^k == 2 (mod k), implying that k is a Fermat pseudoprime (A001567).

Every Super-Poulet number belongs to this sequence.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert Israel)

Maple

filter:= proc(n) local D, d;
  if isprime(n) then return false fi;
  D:= sort(convert(numtheory:-divisors(n), list));
  for d in D while d^2 < n do
    if 2 &^ d - 2 &^(n/d) mod n <> 0 then return false fi
  od:
  true
end proc:
select(filter, [seq(i, i=3..2*10^5, 2)]); # Robert Israel, Aug 28 2017

Mathematica

filterQ[n_] := CompositeQ[n] && AllTrue[Divisors[n], PowerMod[2, #, n] == PowerMod[2, n/#, n]&];
Select[Range[1, 10^6, 2], filterQ] (* Jean-François Alcover, Jun 18 2020 *)

Prog

(PARI) is(k) = {if(k == 1 || !(k%2) || isprime(k), return(0)); fordiv(k, d, if(d^2 <= k && Mod(2, k)^d != Mod(2, k)^(k/d), return(0))); 1; } \\ Amiram Eldar, Apr 22 2024

Crossrefs

Subsequence of A001567.

Supersequence of A050217, their set difference is given by A291602.

Cf. A291602.

Sequence in context: A087835 A271221 A066488 * A367319 A083876 A348258

Adjacent sequences: A291598 A291599 A291600 * A291602 A291603 A291604

Keyword

nonn,changed

Author

Max Alekseyev, Aug 27 2017

Status

approved