A329196 - OEIS

%I #66 Jan 01 2024 22:47:16

%S 10891,12709,11130,11107,11090,43600,44960,45496,44343,44232,44021,

%T 74780,78098,76207,75800,78180,79958,77915,78199,79979,82001,110891,

%U 112709,111130,111107,111090,180164,258316,224791,227119,232727,221172,220107,217990,201781

%N Irregular table whose rows are the nontrivial cycles of the ghost iteration A329200, ordered by increasing smallest member, always listed first.

%C A329200 consists of adding the number whose digits are the absoute values of differences of adjacent digits of n in case it is even, or subtracting it if it is odd. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence lists these cycles, ordered by their smallest member which is always listed first. Sequence A329197 gives the row lengths.

%C Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the (cyclic) first differences are also the same and add again up to 0.) This is the case for rows 1, 2, 3, ... (but not row 4 or 6) of this table. Rows 5, 7 and 8 are the second members of these three families. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.

%H Lars Blomberg, <a href="/A329196/b329196.txt">Table of n, a(n) for n = 1..235 (cycles found for n < 10^10)</a>

%e The table starts:

%e    n |  cycle #n  (length = A329197(n))

%e   ---+-----------------------------------------------------------------------

%e    1 |  10891,  12709,  11130,  11107,  11090

%e    2 |  43600,  44960,  45496,  44343,  44232,  44021

%e    3 |  74780,  78098,  76207

%e    4 |  75800,  78180,  79958,  77915,  78199,  79979, 82001

%e    5 | 110891, 112709, 111130, 111107, 111090

%e    6 | 180164, 258316, 224791, 227119, 232727, 221172, 220107, 217990, 201781

%e    7 | 443600, 444960, 445496, 444343, 444232, 444021

%e    8 | 774780, 778098, 776207

%e    9 | 858699, 891929, 873052

%e   10 | 1110891, 1112709, 1111130, 1111107, 1111090

%e   11 | 3270071, 3427147, 3301514

%e   12 | 4381182, 4538258, 4412625

%e   13 | 4443600, 4444960, 4445496, 4444343, 4444232, 4444021

%e   14 | 5492293, 5649369, 5523736

%e   15 | 7774780, 7778098, 7776207

%e   16 | 8858699, 8891929, 8873052

%e   17 | 11110891, 11112709, 11111130, 11111107, 11111090

%e   18 | 33270071, 33427147, 33301514

%e   19 | 44381182, 44538258, 44412625

%e   20 | 44443600, 44444960, 44445496, 44444343, 44444232, 44444021

%e   21 | 55492293, 55649369, 55523736

%e   22 | 77774780, 77778098, 77776207

%e   23 | 85869922, 89192992, 87305285

%e   24 | 88858699, 88891929, 88873052

%e   25 | 111110891, 111112709, 111111130, 111111107, 111111090

%e   26 | 333270071, 333427147, 333301514

%e   27 | 444381182, 444538258, 444412625

%e   28 | 444443600, 444444960, 444445496, 444444343, 444444232, 444444021

%e   29 | 555492293, 555649369, 555523736

%e   30 | 615930235, 670393447, 653027344, 665352754, 664129233, 666446943,

%e      | 666244592, 665824445, 664462444, 666486644, 666728664, 666884866,

%e      | 667089286, 668871048, 670887192, 653085505, 640702450

%e   31 | 777774780, 777778098, 777776207

%e   32 | 809513051, 898955405, 887815260, 888989606, 889100972, 887290047,

%e      | 885711004, 888971108, 889097126, 891089740, 909270974

%e   33 | 858699257, 891929989, 873052978

%e   34 | 885869922, 889192992, 887305285

%e   35 | 888858699, 888891929, 888873052

%e   36 | 1111110891, 1111112709, 1111111130, 1111111107, 1111111090

%e   37 | 3333270071, 3333427147, 3333301514

%e   38 | 4444381182, 4444538258, 4444412625

%e   39 | 4444443600, 4444444960, 4444445496, 4444444343, 4444444232, 4444444021

%e   40 | 5461740619, 5587375277, 5618817627, 5461741482, 5587374828, 5618818294

%e   41 | 5555492293, 5555649369, 5555523736

%e   42 | 6615930235, 6670393447, 6653027344, 6665352754, 6664129233,

%e      | 6666446943, 6666244592, 6665824445, 6664462444, 6666486644,

%e      | 6666728664, 6666884866,

%e      | 6667089286, 6668871048, 6670887192, 6653085505, 6640702450

%e   43 | 7777774780, 7777778098, 7777776207

%e   44 | 8858699257, 8891929989, 8873052978

%e   45 | 8885869922, 8889192992, 8887305285

%e   46 | 8888858699, 8888891929, 8888873052

%e   47 | 11111110891, 11111112709, 11111111130, 11111111107, 11111111090

%e   48 | 31128941171, 33145094237, 33376689451, 33417710965, 33281649034,

%e      | 33114123103, 32910811890

%e   49 | 44444443600, 44444444960, 44444445496, 44444444343,

%e      | 44444444232, 44444444021

%e The smallest starting value for which the trajectory under A329200 does not end in a fixed point is n = 8059: This leads into a cycle of length 5, 11090 -> 10891 -> 12709 -> 11130 -> 11107 -> 11090. "Rotated" as to start with the smallest member, this yields the first row of this table, (10891, 12709, 11130, 11107, 11090).

%e Starting value n = 37908 leads after two steps into the next cycle (44232, 44021, 43600, 44960, 45496, 44343), of length 6. Again "rotating" this list as to start with the smallest member, it yields the second row of this table.

%e Starting value n = 68060 leads after 8 steps into a new cycle of length 7, (75800, 78180, 79958, 77915, 78199, 79979, 82001). However, this will NOT give row 3 but only row 4, because:

%e The starting value 70502 leads after 3 steps into the loop (74780, 78098, 76207) which comes lexicographically earlier than the previously mentioned cycle of length 7. Therefore this is row 3 of this table.

%e Starting value 70515 enters the loop (111090, 110891, 112709, 111130, 111107) after 15 steps. This becomes row 5.

%e This row 5 is the same as row 1 with the initial digit 1 duplicated in each term: it is the second member of this infinite family of cycles of length 5. Similarly, rows 2 and 3 (where all terms have the same length and initial digit) also lead to infinite families of cycles by successively duplicating the initial digit of each term.

%e The pattern 858699257(257|857)*84302(302|342)* also yields cycles. - _Lars Blomberg_, Nov 15 2019

%o (PARI)

%o T(n,T=[n])={while(!setsearch(Set(T),n=A329200(n)), T=concat(T,n));T} /* trajectory; is a cycle when n is a member of it */

%o {U=0; M=[]; for(n=9,oo, bittest(U>>=1,0) && next; if(M && n>M[1], print(T(M[1])); M=M[^1]); t=n; V=U; while( !bittest(U,-n+t=A329200(t)), t>n || next(2); U+=1<<(t-n)); bittest(V,t-n) || #Set(digits(t))==1 || M=setunion(M,[vecmin(T(t))]) )}

%Y Cf. A329197 (row lengths), A329200, A329198.

%Y Cf. A329342 (analog for the variant A329201).

%K nonn,tabf

%O 1,1

%A _M. F. Hasler_, Nov 10 2019

%E Rows 9 through 35 from _Scott R. Shannon_, Nov 12 2019

%E Table of cycles extended by _Lars Blomberg_, Nov 15 2019