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

%I A122988 #18 Feb 16 2025 08:33:02
%S A122988 1,1000,159,505,52,105,102,505,52,505,22,505,52,505,102,105,52,505,
%T A122988 102,505,12,505,102,505,52,25,102,505,52,505,22,505,52,505,102,105,52,
%U A122988 505,102,505,12,505,102,505,52,105,102,505,52,505,6,505,52,505,102,105,52
%N A122988 Number of possible arrangements of the last three digits of x^n for all x>0 (leading zeros omitted).
%C A122988 Only possible values are {1, 4, 6, 12, 22, 25, 52, 102, 105, 159, 505, 1000}. - _Robert G. Wilson v_, Sep 27 2006.
%H A122988 Robert G. Wilson v, <a href="/A122988/b122988.txt">Table of n, a(n) for n = 0..1000</a>
%H A122988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>.
%H A122988 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquareNumber.html">Square Number</a>.
%F A122988 a(n)=1 for n=0 only,
%F A122988 a(n)=4 for n=100*k, k>=1,
%F A122988 a(n)=6 for n=100*k-50, k>=1,
%F A122988 a(n)=12 for n=20*k, k>=1 except if k == 0 (mod 5),
%F A122988 a(n)=22 for n=20*k-10, k>=1 except if k == 3 (mod 5),
%F A122988 a(n)=25 for n=50*k-25, k>=1,
%F A122988 a(n)=52 for n=4*k, k>=1 except if k == 0 (mod 5),
%F A122988 a(n)=102 for n=4*k-2, k>=2 except if k == 3 (mod 5),
%F A122988 a(n)=105 for n=10*k-5, k>=1 except if k == 3 (mod 5),
%F A122988 a(n)=159 for n=2 only,
%F A122988 a(n)=505 for n=2*k-1, k>=2 except if k == 3 (mod 5) and
%F A122988 a(n)=1000 for n=1 only.
%e A122988 a(0) = 1 because the last three digits of x^0 are always 001 (just one possibility).
%e A122988 a(100)=4 because the last three digits of x^100 can be 000, 001, 376 or 625 (which is four possibilities).
%t A122988 f[n_] := Length@ Union@ PowerMod[ Range@1000, n, 1000]; Table[ f@n, {n, 0, 56}] (* _Robert G. Wilson v_ *)
%Y A122988 Cf. A122986, A122987, A000578, A006716, A027676.
%K A122988 base,nonn
%O A122988 0,2
%A A122988 _Sergio Pimentel_, Sep 22 2006
%E A122988 Edited and extended by _Robert G. Wilson v_, Sep 27 2006