A376506 Natural numbers whose iterated squaring modulo 100 eventually settles at the attractor 1.

1, 7, 43, 49, 51, 57, 93, 99, 101, 107, 143, 149, 151, 157, 193, 199, 201, 207, 243, 249, 251, 257, 293, 299, 301, 307, 343, 349, 351, 357, 393, 399, 401, 407, 443, 449, 451, 457, 493, 499, 501, 507, 543, 549, 551, 557, 593, 599, 601, 607, 643, 649, 651, 657

Offset: 1

Author

Martin Renner, Sep 25 2024

Keywords

• nonn
• easy

Comments

The natural numbers decompose into six categories under the operation of repeated squaring modulo 100, four of which consist of numbers that eventually settle at the attractors 0 (cf. A008592), 1 (this sequence), 25 (cf. A017329), or 76 (cf. A376507), and two of which eventually enter one of the 4-cycles 16, 56, 36, 96 (cf. A376508) or 21, 41, 81, 61 (cf. A376509).

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

Examples

			7^2 = 49 -> 49^2 = 1 -> 1^2 = 1 -> ... (mod 100).
		

References

• Alexander K. Dewdney, Computer-Kurzweil. Mit einem Computer-Mikroskop untersuchen wir ein Objekt von faszinierender Struktur in der Ebene der komplexen Zahlen. In: Spektrum der Wissenschaft, Oct 1985, p. 8-14, here p. 11-13 (Iterations on a finite set), 14 (Iteration diagram).

Links

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

Crossrefs

Cf. A008592, A017329, A376507, A376508, A376509.

Formula

G.f.: x*(1 + 6*x + 36*x^2 + 6*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3)). - Stefano Spezia, Sep 26 2024