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A248218
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Period in residues modulo n in iteration of x^2 + 1 starting at 0.
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16
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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, 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, 6, 2, 6, 11, 6, 6, 2, 3, 2, 2, 4, 4, 6, 9, 14, 5, 2, 6
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OFFSET
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1,2
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COMMENTS
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a(n) is a period in the sequence A003095 modulo n.
For n <= 10000 is the maximal period a(7921) = 1232.
For n <= 100000 is the maximal period a(73205) = 7260.
For n <= 500000 is the maximal period a(357911) = 54670.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 1..100000
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FORMULA
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a(LCM(i,j)) = LCM(a(i),a(j)). - Robert Israel, Mar 08 2021
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EXAMPLE
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n=5, residues are 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, ... , period is 3, a(5)=3.
n=7, residues are 1, 2, 5, 5, 5, 5, 5, ... , final period is 1, therefore a(7)=1.
n=10, residues are 1, 2, 5, 6, 7, 0, 1, 2, 5, 6, 7, 0, 1, 2, ... , a(10)=6.
n=43, residues are 1, 2, 5, 26, 32, 36, 7, 7, 7, 7, ... , a(43) = 1.
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.
This program is for experiments (n<100): Rest[NestList[Mod[#^2+1, n] &, 0, 100]]
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MATHEMATICA
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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}]
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PROG
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(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
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CROSSREFS
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Cf. A248219, A256342 - A256349, A003095, A247981, A001175.
Sequence in context: A008342 A277214 A278603 * A182110 A175328 A338776
Adjacent sequences: A248215 A248216 A248217 * A248219 A248220 A248221
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KEYWORD
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nonn
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AUTHOR
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Vaclav Kotesovec, Oct 04 2014
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STATUS
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approved
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