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

3, 7, 11, 13, 19, 29, 31, 37, 71, 127, 379, 491, 2047, 2633, 2659, 3373, 8191, 13249, 26893, 70687, 74597, 87211, 131071, 184511, 524287, 642581, 1897121, 2676301, 2703739, 8388607, 15456151, 52368101, 102785339, 126233057, 193481677, 536870911, 856645921, 1552107133, 2001907169, 2147483647, 2935442621, 3668158729, 6004262437

Offset: 1

Author

Alzhekeyev Ascar M, Feb 25 2011

Keywords

• nonn

Comments

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

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.

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.

The sequence contains A125854 as a subsequence.

Crossrefs

Cf. A001348, A001567, A125854.

Programs

Maple

isA186645 := proc(n)
        if Power(2,n-1) mod n = 1 then
                x := Power(2,n-1) mod (n^2) ;
                b := (x-1)/n ;
                if b>0 then if modp(n-1,b) = 0 then true; else false; end if;
                else false;
                end if;
        else
                false;
        end if;
end proc:
for n from 1 do if isA186645(n) then printf("%d,\n",n); end if; end do: # R. J. Mathar, Mar 09 2011

Extensions

Edited and more terms added by Max Alekseyev, Mar 14 2011