A307797 -id:A307797 - cp's OEIS frontend

A307536 Self referencing version of the "Kimberling shuffle" sequence (see Comments).

1, 2, 2, 4, 2, 6, 6, 8, 2, 2, 11, 2, 13, 14, 6, 6, 2, 11, 19, 2, 21, 6, 2, 2, 2, 26, 27, 6, 11, 26, 13, 11, 19, 19, 11, 2, 26, 26, 13, 40, 26, 2, 2, 13, 45, 2, 26, 19, 49, 50, 51, 51, 21, 13, 26, 2, 57, 26, 6, 13, 2, 27, 63, 57, 26, 6, 21, 26, 21, 11, 26, 40, 73, 74, 45, 11, 77, 78, 2, 80, 6, 49, 2, 2, 85, 73, 87, 27, 89

Offset: 1

David James Sycamore, Apr 12 2019

nonn

If the first row of the expulsion array is replaced by this sequence, and the rows are "shuffled" then the sequence reappears in the diagonal.
For integer n >= 1 define the set [n]={x; A^r(x)=n}U{y; B^r(y)=n}; (r=0,1,2,3..., A^0(n)=B^0(n)=n), where A=A007063 and B=A006852 (mutual inverses). This set includes n, together with all numbers linked to n by A and B. If a number m is in [n], then [m]=[n], therefore we name the set by its least element k, which takes the following values: 1,2,4,6,8,11,13,14,19,21,26,27,40,45,48,50,51,57,63,... Assuming every n is a term in A, the collection of distinct sets [k] is a partition of the natural numbers, and this sequence is constructed by replacing in the first row of the original array, every number n in [k], with k.
A lexicographically earliest version can be obtained from this sequence by replacing any term > all preceding terms by k+1, where k is the greatest term seen so far. Thus: 1,2,2,3,2,4,4,5,2,2,6,2,7,8,4,4,2,6,9,2,10,4,2,2,2,11,...
From Lars Blomberg, Apr 27 2019: (Start)
Starting with some k value and extending in both directions using A and B results in a "valley" with k at the bottom and often sub-valleys on the hillsides (larger than k). (See the document referenced in A038807 for an illustration.)
So the k sequence is computed by selecting the smallest value not yet seen and iterate as far as possible, then select the next value not seen, etc.
However, while it seems that A and B values goes toward infinity, it is not known whether a known valley will eventually connect to another known valley, leading to a different set of k values.
The DATA is based on iterating A and B until the value > 10^8. (End)

			Examples of [k] for the above list up to k=27:
[1]={1}; so a(1)=1
[2]={2,3,5,9,10,12,17,20,23,24,25,36,42,43,...}; so a(3)=a(5)=a(9)=...=a(43)=2, etc.
[4]={4}; a(4)=4
[6]={6,7,15,16,22,28,59,66,81,...}; a(6)=a(7)=a(15)=...a(81)=6, etc.
[8]={8}; a(8)=8
[11]={11,18,29,32,35,70,76,...}; a(18)=a(29)=...=a(76)=11, etc.
[13]={13,31,39,44,54,60,90,...}; a(31)=a(39)=...=a(90)=13, etc.
[14]={14}; a(14)=14
[19]={19,33,34,48,...}
[21]={21,53,67,69,...}
[26]={26,30,37,38,41,47,55,58,65,68,71,95,99,...}
[27]={27,62,88,...}

R. K. Guy, Unsolved Problems Number Theory, Sect E35.

D. Gale, Tracking the Automatic Ant: And Other Mathematical Explorations, ch. 5, p. 27. Springer, 1998.
C. Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991 and Vol. 18, March 1992, pp. 82-83.
Eric Weisstein's World of Mathematics, Kimberling Sequence

Cf. A007063, A006852, A038807, A307797.

{A(z) = x=z; y=z; xx=2*x-4; while (y<=xx, x--; xx-=2; if (bittest(y,0)==1, y=x+((y+1)>>1), y=x-(y>>1))); return(x+y-1);};
{B(z) = a=z; n=1; while (a!=n, if (a2*n, a--, a=2*(a-n)-1);n++); return(a);}; \\ Lars Blomberg, Apr 29 2019

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A307536 Self referencing version of the "Kimberling shuffle" sequence (see Comments).

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