A376538 Natural numbers whose iterated squaring modulo 1000 eventually settles at the attractor 1.

1, 57, 193, 249, 251, 307, 443, 499, 501, 557, 693, 749, 751, 807, 943, 999, 1001, 1057, 1193, 1249, 1251, 1307, 1443, 1499, 1501, 1557, 1693, 1749, 1751, 1807, 1943, 1999, 2001, 2057, 2193, 2249, 2251, 2307, 2443, 2499, 2501, 2557, 2693, 2749, 2751, 2807

Offset: 1

Author

Martin Renner, Sep 26 2024

Keywords

• nonn
• easy

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 (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).

The first-order differences of the numbers in this sequence repeat with a fixed period of length four: 56, 136, 56, 2, ...

Examples

			57^2 = 249 -> 249^2 = 1 -> 1^2 = 1 -> ... (mod 1000).
		

Links

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

Crossrefs

Cf. A008592, A017329, A376506, A376507, A376508, A376509, A376539, A376540, A376541.