A227359 Natural numbers that are not of the form (k +- sum of binary digits of k) for any k.

6, 13, 21, 30, 37, 48, 51, 80, 83, 111, 121, 133, 144, 147, 175, 185, 192, 207, 217, 226, 233, 242, 245, 248, 250, 272, 275, 303, 313, 320, 335, 345, 354, 361, 370, 373, 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

Offset: 1

Author

Andres M. Torres, Jul 08 2013

Keywords

• nonn
• base

Comments

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.

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

Examples

			Find the list of values not defined by:
V = i +- count of set binary bits(i), for any integer i.
Assume that setbits(n) returns the count of set binary digits of n.
A227359 sample: 6,13,21,30,37,48,51,80,83,111, ...
0 +- setbits(0) = 0     therefore 0 does not make the list
1 +- setbits(1) = 0,2   therefore 0 and 2 do not make the list
2 +- setbits(2) = 1,3   therefore 1 and 3 do not make the list
3 +- setbits(3) = 1,5   therefore 1 and 5 do not make the list
4 +- setbits(4) = 3,5   ...
5 +- setbits(5) = 3,7   therefore 3 and 7 do not make the list
6 +- setbits(6) = 4,8   therefore 4 and 8 do not make the list
7 +- setbits(7) = 4,10  therefore 4 and 10 do not make the list
8 +- setbits(8) = 7,9   therefore 7 and 9 do not make the list
6 and 13 did make the list because there is no solution for
6 = i +- setbits(i),  nor
13 = i +- setbits(i), for any integer i.
		

Links

• Andres M. Torres, Table of n, a(n) for n = 1..1000
• Andres M. Torres, Zip file containing Blitz3D code.

Crossrefs

Cf. A055938, A010061, A010062, A242403.

Programs

Blitz3D

;; See link.