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

1,1

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

Links

Andres M. Torres, Table of n, a(n) for n = 1..1000

Andres M. Torres, Zip file containing Blitz3D code.

Example

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.

Prog

See link.

Crossrefs

Cf. A055938, A010061, A010062, A242403.

Sequence in context: A017053 A046040 A330177 * A056115 A173358 A101247

Adjacent sequences: A227356 A227357 A227358 * A227360 A227361 A227362

Keyword

nonn,base

Author

Andres M. Torres, Jul 08 2013

Status

approved