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A220025 The period with which the powers of n repeat mod 100000. 1
1, 1, 2500, 5000, 1250, 8, 625, 500, 2500, 2500, 1, 5000, 2500, 5000, 1250, 2, 625, 2500, 500, 5000, 1, 5000, 2500, 2500, 250, 4, 125, 5000, 2500, 5000, 1, 1250, 500, 2500, 1250, 8, 625, 5000, 2500, 2500, 1, 2500, 2500, 1000, 1250, 8, 625, 2500, 2500, 250, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) will always be a divisor of Phi(100000) = 40000.

This sequence is periodic with a period of 100000 because n^i mod 100000 = (n + 100000)^i mod 100000.

For the odd numbers n ending in {1, 3, 7, 9} which are coprime to 10, we can expect the powers of n mod 100000 to loop back to 1, with the value of n^a(n) mod 100000 = 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 100000 = 9376.

For the numbers n ending in 5, n^(8*i) mod 100000 = 90625, for all i >= 1.

For the numbers n ending in 0, n^i mod 100000 = 0, for all i >= 5.

LINKS

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

EXAMPLE

a(2) = 2500 since 2^i mod 100000 = 2^(i + 2500) mod 100000, for all i >= 5.

a(3) = 5000 since 3^i mod 100000 = 3^(i + 5000) mod 100000, for all i >= 0.

But a(7) = 500 since 7^i mod 100000 = 7^(i + 500) mod 100000, for all i >= 0.

MATHEMATICA

Flatten[Table[s = Table[PowerMod[n, e, 100000], {e, 2, 100000}]; Union[Differences[Position[s, s[[4]]]]], {n, 0, 40}]] (* Vincenzo Librandi, Jan 26 2013 *)

PROG

(PARI) k=100000; for(n=0, 100, x=(n^5)%k; y=(n^6)%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: A131523 A307764 A062120 * A253377 A253370 A204225

Adjacent sequences:  A220022 A220023 A220024 * A220026 A220027 A220028

KEYWORD

nonn,base

AUTHOR

V. Raman, Dec 15 2012

STATUS

approved

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Last modified October 15 04:33 EDT 2019. Contains 328026 sequences. (Running on oeis4.)