|
%I #18 Sep 08 2022 08:45:01
%S 1,57,68,124,126,182,193,249,251,307,318,374,376,432,443,499,501,557,
%T 568,624,626,682,693,749,751,807,818,874,876,932,943,999,1001,1057,
%U 1068,1124,1126,1182,1193,1249,1251,1307,1318,1374,1376,1432,1443,1499,1501
%N Numbers k such that k^4 == 1 (mod 5^3).
%C Numbers that are congruent to {1, 57, 68, 124} mod 125.
%H Amiram Eldar, <a href="/A056082/b056082.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F From _Wesley Ivan Hurt_, Jun 07 2016: (Start)
%F G.f.: x*(1+56*x+11*x^2+56*x^3+x^4)/((x-1)^2*(1+x+x^2+x^3)).
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
%F a(n) = (250*n-125+99*i^(2*n)+(9-9*i)*i^(-n)+(9+9*i)*i^n)/8 where i=sqrt(-1). (End)
%p A056082:=n->(250*n-125+99*I^(2*n)+(9-9*I)*I^(-n)+(9+9*I)*I^n)/8: seq(A056082(n), n=1..100); # _Wesley Ivan Hurt_, Jun 07 2016
%t x=5; Select[ Range[ 1000 ], PowerMod[ #, x-1, x^3 ]==1& ]
%o (Magma) [n : n in [0..2000] | n mod 125 in [1, 57, 68, 124]]; // _Wesley Ivan Hurt_, Jun 07 2016
%Y Cf. A056083, A056084, A056085, A056086, A056087, A056088, A056089.
%K nonn,easy
%O 1,2
%A _Robert G. Wilson v_, Jun 08 2000
|
