A220025 The period with which the powers of n repeat mod 100000.

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

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