%I #7 Sep 30 2024 15:00:39
%S 7,43,49,51,93,99,101,107,143,149,151,157,199,201,207,243,257,293,299,
%T 301,343,349,351,357,393,399,401,407,449,451,457,493,507,543,549,551,
%U 593,599,601,607,643,649,651,657,699,701,707,743,757,793,799,801,843
%N Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 201, 401, 801, 601.
%C The natural numbers decompose into eight categories under the operation of repeated squaring modulo 1000, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (cf. A376538), 376 (cf. A376539), or 625 (cf. A017329), two of which eventually enter one of the 4-cycles 176, 976, 576, 776 (cf. A376540) or 201, 401, 801, 601 (this sequence), and two of which eventually enter one of the 20-cycles 16, 256, 536, 296, 616, 456, 936, 96, 216, 656, 336, 896, 816, 856, 736, 696, 416, 56, 136, 496 (cf. A376508) or 41, 681, 761, 121, 641, 881, 161, 921, 241, 81, 561, 721, 841, 281, 961, 521, 441, 481, 361, 321 (cf. A376509).
%C The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 36, 6, 2, 42, 6, 2, 6, 36, 6, 2, 6, 42, 2, 6, 36, 14, ...
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).
%e 7^2 = 49 -> 49^2 = 401 -> 401^2 = 801 -> 801^2 = 601 -> 601^2 = 201 -> 201^2 = 401 -> ... (mod 1000).
%Y Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376538, A376539, A376540.
%K nonn
%O 1,1
%A _Martin Renner_, Sep 26 2024