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A299635 Solution (b(n)) of the system of 3 complementary equations in Comments. 3
2, 5, 6, 7, 8, 11, 13, 15, 18, 20, 21, 24, 25, 27, 29, 31, 32, 33, 34, 35, 38, 39, 43, 44, 45, 46, 47, 51, 52, 53, 55, 56, 59, 60, 63, 65, 69, 71, 72, 73, 75, 79, 80, 81, 82, 84, 85, 87, 91, 93, 96, 98, 99, 100, 105, 107, 109, 111, 113, 114, 116, 117, 119 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Define sequences a(n), b(n), c(n) recursively, starting with a(0) = 1, b(0) = 2:

a(n) = least new k >= 2*b(n-1);

b(n) = least new k;

c(n) = a(n) + b(n);

where "least new k" means the least positive integer not yet placed.

***

The sequences a,b,c partition the positive integers.

***

Let x = 11/6.  Conjectures:

a(n) - 2*n*x = 0 for infinitely many n;

b(n) - n*x = 0 for infinitely many n;

c(n) - 3*n*x = 0 for infinitely many n;

(a(n) - 2*n*x) is unbounded below and above;

(b(n) - n*x) is unbounded below and above;

(c(n) - 3*n*x) is unbounded below and above;

***

Let d(a), d(b), d(c) denote the respective difference sequences. Conjectures:

12 occurs infinitely many times in d(a); 6 occurs infinitely many times in d(b);

2 occurs infinitely many times in d(c).

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..1000

EXAMPLE

n:   0    1    2    3    4    5    6    7    8    9

a:   1    4   10   12   14   17   23   26   30   37

b:   2    5    6    7    8   11   13   15   18   20

c:   3    9   16   19   22   28   36   41   48   57

MATHEMATICA

z = 1000;

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);

a = {1}; b = {2}; c = {}; AppendTo[c, Last[a] + Last[b]];

Do[{

  AppendTo[a, mex[Flatten[{a, b, c}], 2 Last[b]]],

  AppendTo[b, mex[Flatten[{a, b, c}], 1]],

  AppendTo[c, Last[a] + Last[b]]}, {z}];

Take[a, 100] (* A299634 *)

Take[b, 100] (* A299635 *)

Take[c, 100] (* A299636 *)

(* Peter J. C. Moses, Apr 08 2018 *)

CROSSREFS

Cf. A299634, A299636.

Sequence in context: A122546 A258775 A275894 * A170944 A074940 A028752

Adjacent sequences:  A299632 A299633 A299634 * A299636 A299637 A299638

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 17 2018

STATUS

approved

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Last modified March 8 20:52 EST 2021. Contains 341953 sequences. (Running on oeis4.)