A220023 The period with which the powers of n repeat mod 1000.

1, 1, 100, 100, 50, 2, 25, 20, 100, 50, 1, 50, 100, 100, 50, 2, 25, 100, 20, 50, 1, 50, 100, 100, 10, 1, 5, 100, 100, 50, 1, 50, 20, 100, 50, 2, 25, 100, 100, 50, 1, 25, 100, 20, 50, 2, 25, 100, 100, 10, 1, 10, 100, 100, 50, 2, 25, 4, 100, 50, 1, 50, 100, 100

Offset

0,3

Comments

a(n) will always be a divisor of Phi(1000) = 400.

This sequence is periodic with a period of 1000 because n^i mod 1000 = (n + 1000)^i mod 1000.

For the odd numbers n ending in {1, 3, 7, 9} which are coprime to 10, we can expect the powers of n mod 1000 to loop back to 1, with the value of n^a(n) mod 1000 = 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 1000 = 376.

For the numbers n ending in 5, n^(2*i) mod 1000 = 625, for all i >= 2.

For the numbers n ending in 0, n^i mod 1000 = 0, for all i >= 3.

Links

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

Example

a(2) = 100 since 2^i mod 1000 = 2^(i+100) mod 1000, for all i >= 3.
a(3) = 100 since 3^i mod 1000 = 3^(i+100) mod 1000, for all i >= 0.
But a(7) = 20 since 7^i mod 1000 = 7^(i+20) mod 1000, for all i >= 0.

Mathematica

Flatten[Table[s=Table[PowerMod[n, e, 1000], {e, 2, 1000}]; Union[Differences[Position[s, s[[2]]]]], {n, 0, 40}]] (* Vincenzo Librandi, Jan 26 2013 *)

Prog

(PARI) k=1000; for(n=0, 100, x=(n^3)%k; y=(n^4)%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: A169735 A096582 A308660 * A115048 A112525 A000865

Adjacent sequences: A220020 A220021 A220022 * A220024 A220025 A220026

Keyword

nonn,base

Author

V. Raman, Dec 15 2012

Status

approved