
COMMENTS

The first 23 terms (at least) are primes.
Conjecture: All terms have the form 12*k+5.
The first composite numbers in the sequence are 2^80+1 and 2^512+1.
If we modify one of the conditions to y  x = 2*n, the sequence changes to 3, 7, 19, 31, 79, 139, 199, 211, 271, 283, 307, 331, 367, 379, 439, 499, 511, ...
or if we modify it to y  x = 64*n, the sequence becomes 89, 101, 197, 269, 317, 341, 461, 521, 569, 701, 821, 857, 881, 929, 1109, 1181, 1217, ...
There seem to be no solutions n if the condition is modified to any y  x <= 0.


MAPLE

isA190638 := proc(n) local b, x, y; b := n*(2*n1) ; x := modp( 2 &^ (b1), b) 1; y := modp( (2*n1) &^ (b1), b) 1; if yx =n and modp(x, n) = 0 and modp(y, n) = 0 then true; else false; end if; end proc:
for n from 2 do if isA190638(n) then print(n); end if; end do: # R. J. Mathar, Jun 04 2011
