%I #14 Feb 16 2021 02:10:58
%S 11293,12139,25399,31261,36199,44869,49471,62521,72397,83086,89737,
%T 91705,98941,124846,125041,134023,138994,144793,164041,166171,170431,
%U 173311,182527,199543,224962,244294,258169,259891,263086,275281,277987
%N Numbers k that are the products of two distinct primes such that 2*k-1, 4*k-3, 8*k-7, 16*k-15, 32*k-31, 64*k-63 and 128*k-127 are also products of two distinct primes.
%e 11293 is a term because 11293 = 23*491, 2*11293 - 1 = 22585 = 5*4517, 4*11293 - 1 = 45169 = 17*2657, 8*11293 - 1 = 90337 = 13*6949, 16*11293 - 1 = 180673 = 79*2287, 32*11293 - 1 = 361345 = 5*72269, 64*11293 - 1 = 722689 = 11*65699, and 128*11293 - 1 = 1445377 = 193*7489.
%t f[n_]:=Last/@FactorInteger[n]=={1,1}; lst={};Do[If[f[n]&&f[2*n-1]&&f[4*n-3]&&f[8*n-7]&&f[16*n-15]&&f[32*n-31]&&f[64*n-63]&&f[128*n-127],AppendTo[lst,n]],{n,11293,4*9!}];lst
%t tdpQ[n_]:=Module[{f=Table[n*2^i-(2^i-1),{i,0,7}]},And@@(Transpose[ FactorInteger[ #]][[2]]=={1,1}&/@f)]; Select[Range[300000],tdpQ] (* _Harvey P. Dale_, Apr 02 2015 *)
%Y Cf. A006881, A177210, A177211, A177212, A177213, A177214, A177215.
%K nonn
%O 1,1
%A _Vladimir Joseph Stephan Orlovsky_, May 04 2010
%E Example moved from Comments field to Example field by _Harvey P. Dale_, Apr 02 2015