%I #28 Oct 02 2022 10:33:13

%S 6,13,21,30,37,48,51,80,83,111,121,133,144,147,175,185,192,207,217,

%T 226,233,242,245,248,250,272,275,303,313,320,335,345,354,361,370,373,

%U 376,378,387,399,409,418,425,434,437,440,442,457,466,469,472,474,481,488,490,497,505,507,528,531,559,569,576,591,601,610,617

%N Natural numbers that are not of the form (k +- sum of binary digits of k) for any k.

%C This sequence is the intersection of sets A010061 and A055938, where: set A010061 is NONE of ( k + count of set binary bits(k) ), and set A055938 is NONE of ( k - count of set binary bits(k) ), for any k.

%C The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 1, 9, 127, 1362, 12921, 128429, 1261747, 12554142, 125697648, 1257065977, ... . Conjecture: This sequence has an asymptotic density (1/2) * A242403 = 0.126330... . - _Amiram Eldar_, Oct 02 2022

%H Andres M. Torres, <a href="/A227359/b227359.txt">Table of n, a(n) for n = 1..1000</a>

%H Andres M. Torres, <a href="https://www.dropbox.com/s/brpxv3tesn25hjs/Sequence%20A227359.zip">Zip file containing Blitz3D code</a>.

%e Find the list of values not defined by:

%e V = i +- count of set binary bits(i), for any integer i.

%e Assume that setbits(n) returns the count of set binary digits of n.

%e A227359 sample: 6,13,21,30,37,48,51,80,83,111, ...

%e 0 +- setbits(0) = 0 therefore 0 does not make the list

%e 1 +- setbits(1) = 0,2 therefore 0 and 2 do not make the list

%e 2 +- setbits(2) = 1,3 therefore 1 and 3 do not make the list

%e 3 +- setbits(3) = 1,5 therefore 1 and 5 do not make the list

%e 4 +- setbits(4) = 3,5 ...

%e 5 +- setbits(5) = 3,7 therefore 3 and 7 do not make the list

%e 6 +- setbits(6) = 4,8 therefore 4 and 8 do not make the list

%e 7 +- setbits(7) = 4,10 therefore 4 and 10 do not make the list

%e 8 +- setbits(8) = 7,9 therefore 7 and 9 do not make the list

%e 6 and 13 did make the list because there is no solution for

%e 6 = i +- setbits(i), nor

%e 13 = i +- setbits(i), for any integer i.

%o See link.

%Y Cf. A055938, A010061, A010062, A242403.

%K nonn,base

%O 1,1

%A _Andres M. Torres_, Jul 08 2013