A336783 -id:A336783 - cp's OEIS frontend

A336744 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly three.

14, 66, 94, 300, 384, 436, 496, 750, 1406, 1794, 2336, 2624, 28034

Offset: 1

Dino Lorenzini, Aug 02 2020

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^2,b}: N to N, with S_{x^2,b}(m) := x_0^2+ ... + x_d^2.
This is the 'sum of the squares of the digits' dynamical system alluded to in the name of the sequence.
It is known that the orbit set {m,S_{x^2,b}(m), S_{x^2,b}(S_{x^2,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^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^2,3).
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. It is natural to wonder whether the set L(x^2,3) is infinite. It is a folklore conjecture that L(x^2,1) = {2,4}.

			For instance, in base 14, the three cycles are (1), (37,85), and (25,122,164,221,123,185,178,244,46). To verify that (37,85) is a cycle in base 14, note that 37=9+2*14, and that 9^2+2^2=85. Similarly, 85=1+6*14, and 1^2+6^2=37.

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.

Cf. A193583, A193585 (where cycles and fixed points are treated separately).
Cf. A336762 (2 cycles).
Cf. A336783 (4 cycles with sum of cubes of the digits).

Integers b such that A193583(b)+A193585(b) = 3. - Michel Marcus, Aug 03 2020

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

Original entry on oeis.org

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

Offset: 1

Makoto Suwama, Aug 03 2020

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).

			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).

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.

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).

cp's OEIS Frontend

A336744 Integers b where the number of cycles under iteration of sum of squares of digits in base b is exactly three.

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