A122988 Number of possible arrangements of the last three digits of x^n for all x>0 (leading zeros omitted).

1, 1000, 159, 505, 52, 105, 102, 505, 52, 505, 22, 505, 52, 505, 102, 105, 52, 505, 102, 505, 12, 505, 102, 505, 52, 25, 102, 505, 52, 505, 22, 505, 52, 505, 102, 105, 52, 505, 102, 505, 12, 505, 102, 505, 52, 105, 102, 505, 52, 505, 6, 505, 52, 505, 102, 105, 52

Offset: 0

Author

Sergio Pimentel, Sep 22 2006

Keywords

• base
• nonn

Comments

Only possible values are {1, 4, 6, 12, 22, 25, 52, 102, 105, 159, 505, 1000}. - Robert G. Wilson v, Sep 27 2006.

Examples

			a(0) = 1 because the last three digits of x^0 are always 001 (just one possibility).
a(100)=4 because the last three digits of x^100 can be 000, 001, 376 or 625 (which is four possibilities).
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Cubic Number.
• Eric Weisstein's World of Mathematics, Square Number.

Crossrefs

Cf. A122986, A122987, A000578, A006716, A027676.

Programs

Mathematica

f[n_] := Length@ Union@ PowerMod[ Range@1000, n, 1000]; Table[ f@n, {n, 0, 56}] (* Robert G. Wilson v *)

Formula

a(n)=1 for n=0 only,

a(n)=4 for n=100*k, k>=1,

a(n)=6 for n=100*k-50, k>=1,

a(n)=12 for n=20*k, k>=1 except if k == 0 (mod 5),

a(n)=22 for n=20*k-10, k>=1 except if k == 3 (mod 5),

a(n)=25 for n=50*k-25, k>=1,

a(n)=52 for n=4*k, k>=1 except if k == 0 (mod 5),

a(n)=102 for n=4*k-2, k>=2 except if k == 3 (mod 5),

a(n)=105 for n=10*k-5, k>=1 except if k == 3 (mod 5),

a(n)=159 for n=2 only,

a(n)=505 for n=2*k-1, k>=2 except if k == 3 (mod 5) and

a(n)=1000 for n=1 only.

Extensions

Edited and extended by Robert G. Wilson v, Sep 27 2006