This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376538 #8 Sep 30 2024 14:57:44 %S A376538 1,57,193,249,251,307,443,499,501,557,693,749,751,807,943,999,1001, %T A376538 1057,1193,1249,1251,1307,1443,1499,1501,1557,1693,1749,1751,1807, %U A376538 1943,1999,2001,2057,2193,2249,2251,2307,2443,2499,2501,2557,2693,2749,2751,2807 %N A376538 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 1. %C A376538 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 (this sequence), 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 (cf. A376541), 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 A376538 The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 56, 136, 56, 2, ... %H A376538 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1, 0, 0, 1, -1). %e A376538 57^2 = 249 -> 249^2 = 1 -> 1^2 = 1 -> ... (mod 1000). %Y A376538 Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376539, A376540, A376541. %K A376538 nonn,easy %O A376538 1,2 %A A376538 _Martin Renner_, Sep 26 2024