A336762 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly two.

6, 10, 16, 20, 26, 40, 8626, 481360

Offset

1,1

Comments

Let b > 1 be an integer, and write the base b expansion of any nonnegative integer n as n = x_0 + x_1 b + ... + x_d b^d with x_d > 0 and 0 <= x_i < b for i=0,...,d.

Consider the map S_{x^2,b}: N to N, with S_{x^2,b}(n) := x_0^2+ ... + x_d^2.

It is known that the orbit set {n, S_{x^2,b}(n), S_{x^2,b}(S_{x^2,b}(n)), ...} is finite for all n > 0. Each orbit contains a finite cycle, and for a given b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^2,i) the set of bases b such that the set of cycles associated to S_{x^2,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^2,2).

A 1978 conjecture of Hasse and Prichett describes the set L(x^2,2). New elements have been added to this set in the paper Integer Dynamics, by D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang. The sequence contains all b <= 10^6 that are in L(x^2,2).

Links

Table of n, a(n) for n=1..8.

H. Hasse and G. Prichett, A conjecture on digital cycles, J. reine angew. Math. 298 (1978), 8--15. Also on GDZ.

D. Lorenzini, M. Melistas, A. Suresh, M. Suwama, and H. Wang, Integer Dynamics, preprint.

Example

For instance, b = 10 is in this sequence since in the decimal system, there are exactly two cycles (1) and (4, 16, 37, 58, 89, 145, 42, 20).

Crossrefs

Cf. A193583 and A193585 (b is in this sequence if A193583(b)+A193585(b) = 2).

Cf. A336744 (3 cycles).

Cf. A336783 (4 cycles with sum of cubes of the digits).

Sequence in context: A315311 A315312 A315313 * A315314 A315315 A315316

Adjacent sequences: A336759 A336760 A336761 * A336763 A336764 A336765

Keyword

nonn,base,hard,more

Author

Makoto Suwama, Aug 03 2020

Status

approved