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 A248218 Period in residues modulo n in iteration of x^2 + 1 starting at 0. 16

%I

%S 1,2,1,2,3,2,1,2,3,6,2,2,4,2,3,2,6,6,1,6,1,2,2,2,3,4,3,2,2,6,1,2,2,6,

%T 3,6,1,2,4,6,7,2,1,2,3,2,4,2,6,6,6,4,2,6,6,2,1,2,3,6,10,2,3,2,12,2,2,

%U 6,2,6,11,6,6,2,3,2,2,4,4,6,9,14,5,2,6

%N Period in residues modulo n in iteration of x^2 + 1 starting at 0.

%C a(n) is a period in the sequence A003095 modulo n.

%C For n <= 10000 is the maximal period a(7921) = 1232.

%C For n <= 100000 is the maximal period a(73205) = 7260.

%C For n <= 500000 is the maximal period a(357911) = 54670.

%H Vaclav Kotesovec, <a href="/A248218/b248218.txt">Table of n, a(n) for n = 1..100000</a>

%F a(LCM(i,j)) = LCM(a(i),a(j)). - _Robert Israel_, Mar 08 2021

%e n=5, residues are 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, ... , period is 3, a(5)=3.

%e n=7, residues are 1, 2, 5, 5, 5, 5, 5, ... , final period is 1, therefore a(7)=1.

%e n=10, residues are 1, 2, 5, 6, 7, 0, 1, 2, 5, 6, 7, 0, 1, 2, ... , a(10)=6.

%e n=43, residues are 1, 2, 5, 26, 32, 36, 7, 7, 7, 7, ... , a(43) = 1.

%e n=229, residues are 1, 2, 5, 26, 219, 101, 126, 76, 52, 186, 18, 96, 57, 44, 105, 34, 12, 145, 187, 162, 139, 86, 69, 182, 149, 218, 122, 0, 1, 2, 5, 26, 219, 101, 126, 76, 52, 186, 18, 96, 57, 44, 105, 34, 12, 145, 187, 162, 139, 86, 69, 182, 149, 218, 122, 0, 1, 2, 5, 26, ... , period is 28, a(229)=28.

%e This program is for experiments (n<100): Rest[NestList[Mod[#^2+1, n] &, 0, 100]]

%t Table[m=Rest[NestList[Mod[#^2+1,n]&,0,1000]]; period=0; j=1; While[j<=Length[m] && period==0,If[m[[Length[m]-j]]==m[[Length[m]]],period=j]; j++]; period,{n,1,1000}]

%o (PARI) A248218(m,t=0,u=[t])=until(#Set(u=concat(u,t=(t^2+1)%m))<#u,);for(i=1,#u,t==u[#u-i]&&return(i)) \\ _M. F. Hasler_, Mar 25 2015

%Y Cf. A248219, A256342 - A256349, A003095, A247981, A001175.

%K nonn

%O 1,2

%A _Vaclav Kotesovec_, Oct 04 2014

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Last modified June 19 13:38 EDT 2021. Contains 345138 sequences. (Running on oeis4.)