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A192734 Smallest number having binary weight of 3 and n distinct prime factors.

%I A192734 #28 Jul 22 2025 12:15:34
%S A192734 7,21,273,16401,1048593,4295032833,1099512676353,9007199256838145,
%T A192734 302231455185132270387201,1208944266358702884257793,
%U A192734 1329227995784915872903807060297121793,1393796574908163946347162983661240005427201
%N A192734 Smallest number having binary weight of 3 and n distinct prime factors.
%C A192734 Written in binary, these numbers have exactly three 1 bits and the other bits are all 0's. This means that these numbers are of the sum of 1 plus two larger distinct powers of 2. - _Alonso del Arte_, Jul 08 2011
%C A192734 a(n) > A002110(n). [_Reinhard Zumkeller_, Jul 09 2011]
%C A192734 Sequence is not monotone: a(12) > a(13). [_Charles R Greathouse IV_, Jul 11 2011]
%H A192734 Charles R Greathouse IV, <a href="/A192734/b192734.txt">Table of n, a(n) for n = 1..14</a>
%t A192734 list = {7}; For[max = 1; n = 2, n < 120,
%t A192734 For[m = 0, m < n,
%t A192734   tal = 2*(2^n + 2^m) + 1; num = PrimeNu[tal];
%t A192734   If[num > max, AppendTo[list, tal]; max = num]
%t A192734   , m++], n++] (* Sarnbratt *)
%t A192734 A084468 = Flatten[Table[2^m + 2^n + 1, {m, 2, 80}, {n, m - 1}]]; Flatten[Table[Take[Select[A084468, PrimeNu[#] == n &], 1], {n, 10}]] (* _Alonso del Arte_, Jul 08 2011 *)
%o A192734 (PARI) a(n)={
%o A192734   my(t);
%o A192734   for(a=2,9e9,
%o A192734     t=1+1<<a;
%o A192734     for(b=1,a-1,
%o A192734       if(omega(t+1<<b)==n,return(t+1<<b))
%o A192734     )
%o A192734   )
%o A192734 }; \\ _Charles R Greathouse IV_, Jul 08 2011
%o A192734 (Haskell)
%o A192734 a192734 n = head [x | x <- [2^u + 2^v + 1 | u <- [2..], v <- [1..u-1]],
%o A192734                       a001221 x == n]
%o A192734 -- _Reinhard Zumkeller_, Jun 14 2015, Jul 09 2011
%Y A192734 Cf. A084468, A001221.
%K A192734 nonn,base
%O A192734 1,1
%A A192734 _Johan Särnbratt_, Jul 08 2011
%E A192734 a(9) corrected by _Charles R Greathouse IV_, Jul 08 2011
%E A192734 a(12) from _Charles R Greathouse IV_, Jul 11 2011