A298758 - OEIS

%I #31 Nov 09 2023 08:52:49

%S 15709,65281,20770621,104484601,112037185,196049701,425967301,

%T 2593182901,16923897871,32548281361,45812984491,52035130951,

%U 55897227751,82907336737,90003640021,92010062101,138016057141,204082130071,310026150211,620006892121,622333751509

%N Numbers k such that both k and 2k-1 are Poulet numbers (Fermat pseudoprimes to base 2).

%C 2*a(n) - 1 = A303531(n) belongs to A217465. - _Max Alekseyev_, Apr 24 2018

%C Numbers k such that both k and 2k+1 are Poulet numbers are listed in A303447.

%C If p is a prime such that 2*p-1 is also a prime (A005382) and k = (2^(2*p-1)+1)/3 and 2*k-1 are both composites, then k is a term of this sequence (Rotkiewicz, 2000). - _Amiram Eldar_, Nov 09 2023

%H Amiram Eldar, <a href="/A298758/b298758.txt">Table of n, a(n) for n = 1..3031</a> (terms below 2^64)

%H Andrzej Rotkiewicz, <a href="https://eudml.org/doc/23839">Arithmetic progressions formed by pseudoprimes</a>, Acta Mathematica et Informatica Universitatis Ostraviensis, Vol. 8, No. 1 (2000), pp. 61-74.

%t s = Import["b001567.txt", "Data"][[All, -1]]; n = Length[s];

%t aQ[n_] := ! PrimeQ[n] && PowerMod[2, (n - 1), n] == 1;

%t a = {}; Do[p = 2*s[[k]] - 1; If[aQ[p], AppendTo[a, s[[k]]]], {k, 1, n}]; a (* using the b-File from A001567 *)

%o (PARI) isP(n) = (Mod(2, n)^n==2) && !isprime(n) && (n>1);

%o isok(n) = isP(n) && isP(2*n-1); \\ _Michel Marcus_, Mar 09 2018

%Y Subsequence of A001567.

%Y Cf. A005382, A217465, A303447, A303448.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jan 26 2018