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

1,2

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, ...

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

Table of n, a(n) for n=1..54.

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

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

Example

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

Crossrefs

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

Sequence in context: A043064 A175159 A173828 * A376541 A045800 A159254

Adjacent sequences: A376503 A376504 A376505 * A376507 A376508 A376509

Keyword

nonn,easy

Author

Martin Renner, Sep 25 2024

Status

approved