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

1,2

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, ...

Links

Table of n, a(n) for n=1..46.

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

Example

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

Crossrefs

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

Sequence in context: A044770 A008877 A338078 * A336191 A277805 A158660

Adjacent sequences: A376535 A376536 A376537 * A376539 A376540 A376541

Keyword

nonn,easy

Author

Martin Renner, Sep 26 2024

Status

approved