easy iteration generates polynomial sequences

　　K-sequence is defined as follows, K is for "kimyo" which means strange in Japanese :

　　　　 x(n)=[a*x(n-1)+b]/p^r,　 a, b are real number, [x] is integer part of x,

　　　　　　　　　　　　　　　　　p is prime, p^r is the highest power of p dividing [a*x(n-1)+b]

　　This is a generalization of Collatz's 3x+1 sequence.

　　Because, K-sequence is the iteration of f(x)=[a*x+b]/p^r, and if p=2 and a=1.5*2^s and b=0.5*2^s+c, 0<=s, 0<=c<1

　　　　　　　 =(1.5*2^s*x+0.5*2^s)/2^r

　　It becomes the definition of Collatz's sequence.

　　But for example 3,10,5,16,8,4,2,1,4,2,1,--- is represented more compactly as　　　　 3,5,1,1,1,1,1---　.

　　3. many polynomial subsequences.　　　Records

　　LLI, an abrreviation of "a lot of linear parts which are isolated each other", is defined as follows :

　　　　 K-sequence such that p=2, a=2+e, 0<e<k, k is about 0.01

　　Indeed, on LLI　{ x0,p,a,b=1,2,2.00013,3.0 }, between 0-th term and 10000-th term, there are 44 linear subsequences such that 4<={number of terms}.

　　　　 x(16) to x(3613)　　　　　x(n)=x(n-1)+2

　　　　 x(3650) to x(3655)　　 x(n)=x(n-1)+1028

　　　　 x(3820) to x(3848)　　　x(n)=x(n-1)+66

　　From a linear part to next linear part, LLI behaves as if it were a random sequence.

　　Besides them, LLI has second degree's parts, and third and fourth degree's parts.

　　LLI　{ x0, p, a, b = 2443499297,2,2.00014,3.0 } :

　　the 68 terms from 0-th to 67-th are represented as follows.

　　 LLI　{ x0,p,a,b=2417903893829791,2,2.00013,3.0 } :

　　the 5 terms from 0-th term to 4-th term are represented as follows.

　　x(n) = 664/6*n^3+5107490*n^2+471475936498/3*n+2417903893829791

　　LLI　{ x0,p,a,b=2015985557869547951,2,2.00013,3.2 }　　

　　a subsequence of 4th degree exists, from 0-th term to 7-th term.

　　2015985557869547951　2016116596930809473　2016247644509609977

　　2016378700606503103　2016509765222042527　2016640838356781961

　　2016771920011275153　2016903010186075887　

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.00001,3.0　　　　period=180492

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.00013,3.2　　　　period=6366

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.00013,3.0　　　　period=1790641

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.0001,3.0　　　　 period=13677

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.001,3.2　　　　　period=1670

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.001,3.0　　　　　period=11078

　　　　　　　　　　　　　　　　 x0,p,a,b=12345,2,2.001,2.8　　　period=11048

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.00001,3.2　　　　 period unknown

　　　　　　　　　　　　　　　　 x0,p,a,b=1,2,2.00014,3.0　　　　 period=73667

　　　　　　　　　　　　　　　　 x0,p,a,b=10^50+1,2,2.001,3.2　　 period unknown

　　o ESS, for extremely strange sequence, is defined as follows :

　　　　 K-sequence such that a=p^r+e , 0<e<k , k is about 0.01.

　　　ESS looks like a mosaic of periodic subsequences and linear subsequences and random behaving subsequence.

　　　Only LLI is different from the other ESS.

　　　For example, ESS, x0,p,a,b=1,2,4.001, 1.2 is similar to ESS, x0,p,a,b=1,3,3.00001,1.2 , but is absolutely different to LLI.

　　o Another class : x0=3, p=2, a=1.6, 1.0<b<=1.2

　　　These sequences seem to be unbounded.

　　　The period of K-sequence such that x0,p,a,b=107,2,1.6,1.1 is still unknown

　　　　　　　Don Reble calculated x(2,000,000) = 852756...564079; it has 46892digits.

　　5. a program for K-sequence in Mathematica

　　g[x_] := If[Mod[Floor[a*x + b], p] == 0,

　　　　　Floor[a*x + b]/p^IntegerExponent[Floor[a*x + b], p], Floor[a*x + b]];

{1, 6, 31, 156, 781, 3906, 19531, 97657, 97659, 97661, 97663, 19533, 97667, \

97669, 97671, 97673, 3907, 19536, 97682, 97684, 97686, 97688, 19538, 97692,\

97694, 97696, 97698, 3908, 19541, 97707, 97709, 97711, 97713, 19543, 97717,\

97719, 97721, 97723, 3909, 19546, 97732, 97734, 97736, 97738, 19548, 97742,\

97744, 97746, 97748, 782, 3911, 19556, 97782, 97784, 97786, 97788, 19558, \

97792, 97794, 97796, 97798, 3912, 19561, 97807, 97809, 97811, 97813, 19563, \

97817, 97819, 97821, 97823, 3913, 19566, 97832, 97834, 97836, 97838, 19568, \

97842, 97844, 97846, 97848, 3914, 19571, 97857, 97859, 97861, 97863, 19573, \

97867, 97869, 97871, 97873, 783, 3916, 19581, 97907, 97909, 97911, 97913, \

19583, 97917, 97919, 97921, 97923, 3917, 19586, 97932, 97934, 97936, 97938, \

19588, 97942, 97944, 97946, 97948, 3918, 19591, 97957, 97959, 97961, 97963, \

19593, 97967, 97969, 97971, 97973, 3919, 19596, 97982, 97984, 97986, 97988, \

19598, 97992, 97994, 97996, 97998, 784, 3921, 19606, 98032, 98034, 98036, \

98038, 19608, 98042, 98044, 98046, 98048, 3922, 19611, 98057, 98059, 98061, \

98063, 19613, 98067, 98069, 98071, 98073, 3923, 19616, 98082, 98084, 98086, \

98088, 19618, 98092, 98094, 98096, 98098, 3924, 19621, 98107, 98109, 98111, \

98113, 19623, 98117, 98119, 98121, 98123, 157, 786, 3931, 19656, 98282, \

98284, 98286, 98288, 19658, 98292, 98294, 98296, 98298, 3932, 19661, 98307, \

　　　　 1. Why such an easy iteration generates many polynomial subsequences?

　　　　 2. Find the periods of sequences whose periods are not known.

　　　　 " For any n, an LLI which has n-th degree polynomial sequences exists. "

　　All mathematicians, please study about this strange sequence !!

　　Neil Sloane once said, "Indeed, it is strange."

　　Eric Roosentaal said, "There are a few ways to generalize Collats, but Ithink you have taken the broadest definition possible."

　　Another mathematician said, " He found all kinds of weird sequences. "

　　Three examples of K-sequence are mentioned on On-line Encyclopedia of Integer Sequences.

　　a graph of the first 1000 terms of ESS { x0,p,a,b = 1,2,4.001,1.2 }　first line represents 0 to 500, and second line represents 501 to 1000. It behaves cyclic like -> chaotic -> linear -> random * exponetial.　　