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A positive integer n is included if n can be added to d(n) in binary without any carries, where d(n) is the number of positive divisors of n. In other words, no 1 in the binary representation of n is in the same position as any 1 in the binary representation of d(n).
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%I #8 Apr 09 2014 10:15:19

%S 4,5,8,10,13,16,17,26,27,29,32,33,34,35,36,37,41,48,51,53,54,57,58,61,

%T 64,65,66,70,73,74,80,82,89,91,96,97,100,101,102,106,109,112,113,114,

%U 115,122,123,128,129,130,135,137,144,145,146,149,153,155,157,160,161

%N A positive integer n is included if n can be added to d(n) in binary without any carries, where d(n) is the number of positive divisors of n. In other words, no 1 in the binary representation of n is in the same position as any 1 in the binary representation of d(n).

%H Harvey P. Dale, <a href="/A140459/b140459.txt">Table of n, a(n) for n = 1..10000</a>

%e 27 in binary is 11011. 27 has 4 divisors. 4 in binary is 100. Comparing 11011 and 100, it is seen that the two do not have any ones in the same position, so 27 is included in the sequence.

%t bcQ[n_]:=Module[{bd=IntegerDigits[n,2],p},p=PadLeft[IntegerDigits[ DivisorSigma[ 0,n],2],Length[bd],0];Union[Pick[bd,p,1]]=={0}]; Select[ Range[200],bcQ] (* _Harvey P. Dale_, Jan 26 2014 *)

%K base,nonn

%O 1,1

%A _Leroy Quet_, Jul 22 2008

%E Extended beyond 32, _R. J. Mathar_, Aug 04 2008