A336783 Integers b where the number of cycles under iteration of sum of cubes of digits in base b is exactly four.

5, 90, 188

Offset: 1

Author

Haiyang Wang, Aug 04 2020

Keywords

• nonn
• base
• hard
• more

Comments

Let b > 1 be an integer, and write the base b expansion of any nonnegative integer m as m = 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^3,b}: N to N, with S_{x^3,b}(m) := x_0^3+ ... + x_d^3.

It is known that the orbit set {m,S_{x^3,b}(m), S_{x^3,b}(S_{x^3,b}(m)), ...} is finite for all m>0. Each orbit contains a finite cycle, and for a given base b, the union of such cycles over all orbit sets is finite. Let us denote by L(x^3,i) the set of bases b such that the set of cycles associated to S_{x^3,b} consists of exactly i elements. In this notation, the sequence is the set of known elements of L(x^3,4).

Meanwhile, the known terms of the sequence L(x^3,1) is {2}, L(x^3,2) is empty, and L(x^3,3) is {3, 26}. It's undetermined whether the complete sequences are finite, if so, whether the above give all terms.

Examples

			For instance, when b=5 the associated four cycles are (1),(28),(118) and (9,65,35).
		

Links

• 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.

Crossrefs

Cf. A194025, A194281.

Cf. A336744 and A336762 (sum of squares of digits).

Formula

Integers b where A194025(b) + A194281(b) = 4.