A291602 -id:A291602 - cp's OEIS frontend

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

Max Alekseyev, Aug 27 2017

nonn

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.

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

Subsequence of A001567.
Supersequence of A050217, their set difference is given by A291602.
Cf. A291602.

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

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 *)

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

A291612 Carmichael numbers k that satisfy 2^d == 2^(k/d) (mod k) for all d|k and are not Super-Poulet numbers (A050217).

Original entry on oeis.org

1105, 852841, 3828001, 17098369, 118901521, 150846961, 172947529, 186393481, 200753281, 686059921, 771043201, 1001152801, 1207252621, 1269295201, 1632785701, 1772267281, 2301745249, 4765950001, 4897161361, 5278692481, 6030849889, 8251854001, 12121569601, 12456671569

Offset: 1

Max Alekseyev, Thomas Ordowski, Altug Alkan, Aug 27 2017

nonn

Intersection of A002997 and A291602.

			Carmichael number 1105 = 5*13*17 is a term because 2^5 == 2^(13*17) (mod 1105), 2^13 == 2^(5*17) (mod 1105), 2^17 == 2^(5*13) (mod 1105) and it is not a Super-Poulet number.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..2665 from Max Alekseyev)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A050217, A291602.

cp's OEIS Frontend

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

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