A376540 Natural numbers whose iterated squaring modulo 1000 eventually enters the 4-cycle 176, 976, 576, 776.

18, 24, 26, 32, 74, 76, 82, 118, 132, 168, 174, 176, 218, 224, 226, 232, 268, 274, 276, 282, 324, 326, 332, 368, 382, 418, 424, 426, 468, 474, 476, 482, 518, 524, 526, 532, 574, 576, 582, 618, 632, 668, 674, 676, 718, 724, 726, 732, 768, 774, 776, 782, 824

Offset: 1

Author

Martin Renner, Sep 26 2024

Keywords

• nonn

Comments

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 (this sequence) 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).

The first-order differences of the numbers in this sequence repeat with a fixed period of length sixteen: 6, 2, 6, 42, 2, 6, 36, 14, 36, 6, 2, 42, 6, 2, 6, 36, ...

Examples

			18^2 = 324 -> 324^2 = 976 -> 976^2 = 576 -> 576^2 = 776 -> 776^2 = 176 -> 176^2 = 976 -> ... (mod 1000).
		

Links

• Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Crossrefs

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376538, A376539, A376541.