A186645 - OEIS

%I #37 Jan 29 2019 23:59:11

%S 3,7,11,13,19,29,31,37,71,127,379,491,2047,2633,2659,3373,8191,13249,

%T 26893,70687,74597,87211,131071,184511,524287,642581,1897121,2676301,

%U 2703739,8388607,15456151,52368101,102785339,126233057,193481677,536870911,856645921,1552107133,2001907169,2147483647,2935442621,3668158729,6004262437

%N Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k-1.

%C All composites in this sequence are 2-pseudoprimes, A001567.

%C The sequence contains all Mersenne numbers, A001348, k=2^p-1 for primes p (for which b=(k-1)/p). Correspondingly, the composites in this sequence contain all terms of A065341.

%C The sequence also contains composites of the form 2^A001567(j) - 1, which do not belong to A065341. The existence of composites in the sequence that are not of the form 2^x-1 is unclear.

%C The sequence contains A125854 as a subsequence.

%p isA186645 := proc(n)

%p         if Power(2,n-1) mod n = 1 then

%p                 x := Power(2,n-1) mod (n^2) ;

%p                 b := (x-1)/n ;

%p                 if b>0 then if modp(n-1,b) = 0 then true; else false; end if;

%p                 else false;

%p                 end if;

%p         else

%p                 false;

%p         end if;

%p end proc:

%p for n from 1 do if isA186645(n) then printf("%d,\n",n); end if; end do: # _R. J. Mathar_, Mar 09 2011

%Y Cf. A001348, A001567, A125854.

%K nonn

%O 1,1

%A _Alzhekeyev Ascar M_, Feb 25 2011

%E Edited and more terms added by _Max Alekseyev_, Mar 14 2011