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A220024 The period with which the powers of n repeat mod 10000. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(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", "))

CROSSREFS

Cf. A173635 (period with which the powers of n repeat mod 10).

Cf. A220022 (period with which the powers of n repeat mod 100).

Sequence in context: A184146 A214191 A132475 * A251813 A045215 A194085

Adjacent sequences:  A220021 A220022 A220023 * A220025 A220026 A220027

KEYWORD

nonn,base

AUTHOR

V. Raman, Dec 15 2012

STATUS

approved

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Last modified April 14 00:07 EDT 2021. Contains 342941 sequences. (Running on oeis4.)