cp's OEIS Frontend

A186645 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k-1.

%I A186645 #37 Jan 29 2019 23:59:11
%S A186645 3,7,11,13,19,29,31,37,71,127,379,491,2047,2633,2659,3373,8191,13249,
%T A186645 26893,70687,74597,87211,131071,184511,524287,642581,1897121,2676301,
%U A186645 2703739,8388607,15456151,52368101,102785339,126233057,193481677,536870911,856645921,1552107133,2001907169,2147483647,2935442621,3668158729,6004262437
%N A186645 Numbers k such that 2^(k-1) == 1 + b*k (mod k^2), where b divides k-1.
%C A186645 All composites in this sequence are 2-pseudoprimes, A001567.
%C A186645 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 A186645 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 A186645 The sequence contains A125854 as a subsequence.
%p A186645 isA186645 := proc(n)
%p A186645         if Power(2,n-1) mod n = 1 then
%p A186645                 x := Power(2,n-1) mod (n^2) ;
%p A186645                 b := (x-1)/n ;
%p A186645                 if b>0 then if modp(n-1,b) = 0 then true; else false; end if;
%p A186645                 else false;
%p A186645                 end if;
%p A186645         else
%p A186645                 false;
%p A186645         end if;
%p A186645 end proc:
%p A186645 for n from 1 do if isA186645(n) then printf("%d,\n",n); end if; end do: # _R. J. Mathar_, Mar 09 2011
%Y A186645 Cf. A001348, A001567, A125854.
%K A186645 nonn
%O A186645 1,1
%A A186645 _Alzhekeyev Ascar M_, Feb 25 2011
%E A186645 Edited and more terms added by _Max Alekseyev_, Mar 14 2011