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A220024
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The period with which the powers of n repeat mod 10000.
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1
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1, 1, 500, 500, 250, 4, 125, 100, 500, 250, 1, 500, 500, 500, 250, 2, 125, 500, 100, 500, 1, 500, 500, 500, 50, 2, 25, 500, 500, 500, 1, 250, 100, 500, 250, 4, 125, 500, 500, 250, 1, 250, 500, 100, 250, 4, 125, 500, 500, 50, 1, 100, 500, 500, 250, 2, 125, 20
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OFFSET
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0,3
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COMMENTS
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a(n) will always be a divisor of Phi(10000) = 4000.
This sequence is periodic with a period of 10000 because n^i mod 10000 = (n + 10000)^i mod 10000.
For the odd numbers n ending in {1, 3, 7, 9} which are coprime to 10, we can expect the powers of n mod 10000 to loop back to 1, with the value of n^a(n) mod 10000 = 1, but for the other numbers n that are not coprime to 10, they do not loop back to 1.
For the even numbers n ending in {2, 4, 6, 8}, n^a(n) mod 10000 = 9376.
For the numbers n ending in 5, n^(4*i) mod 10000 = 625, for all i >= 1.
For the numbers n ending in 0, n^i mod 10000 = 0, for all i >= 4.
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LINKS
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EXAMPLE
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a(2) = 500 since 2^i mod 10000 = 2^(i + 500) mod 10000, for all i >= 4.
a(3) = 500 since 3^i mod 10000 = 3^(i + 500) mod 10000, for all i >= 0.
But a(7) = 100 since 7^i mod 10000 = 7^(i + 100) mod 10000, for all i >= 0.
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MATHEMATICA
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Flatten[Table[s = Table[PowerMod[n, e, 10000], {e, 2, 10000}]; Union[Differences[Position[s, s[[3]]]]], {n, 0, 40}]] (* Vincenzo Librandi, Jan 26 2013 *)
Table[Length[FindTransientRepeat[PowerMod[n, Range[3000], 10000], 3] [[2]]], {n, 0, 60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 08 2016 *)
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PROG
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(PARI) k=10000; for(n=0, 100, x=(n^4)%k; y=(n^5)%k; z=1; while(x!=y, x=(x*n)%k; y=(y*n*n)%k; z++); print1(z", "))
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CROSSREFS
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Cf. A173635 (period with which the powers of n repeat mod 10).
Cf. A220022 (period with which the powers of n repeat mod 100).
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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