Double Iterations Involving Ceiling and Floor Functions

On May 21st 2022, I created a post titled Iterations of a Ceiling Function and the function being dealt with had this form:$$\lceil \sqrt{n} \, \rceil \times (\lceil \sqrt{n}\, \rceil^2 – n) $$The iteration, with output becoming the new input, leads to zero or a loop although this hasn't been rigorously proved. The number associated with my diurnal age today, 26872, has this property. The trajectory ends in a loop and is as follows:

26872, 3936, 2079, 1702, 2604, 5200, 9417, 18326, 23120, 44217, 64144, 94488, 115808, 161293, 125022, 104076, 81719, 22022, 26671, 36900, 67357, 63180, 81648, 42328, 22248, 37800, 43875, 47250, 59732, 71785, 10452, 16171, 27264, 48472, 81549, 70642, 30324, 52675, 51750, 53352, 2079

Figure 1 shows the trajectory of 26872 under this iteration.

Figure 1: trajectory of 26872 under the floor iteration

There is an associated function that involves the floor function and it has this form:$$\lfloor \sqrt{n} \, \rfloor \times (n-\lfloor \sqrt{n}\, \rfloor^2) $$When this function is iterated, with output serving as the new input, a loop is also reached:

26872, 49389, 23310, 31312, 59136, 21141, 16820, 23091, 43790, 22781, 42150, 25625, 4000, 1953, 748, 513, 638, 325, 18, 8, 8

Figure 2 shows the trajectory of 26872 under the floor iteration.

Figure 2: trajectory of 26872 under ceiling iteration

Here is a permalink to an algorithm that will confirm these two trajectories. Numbers like 26872 that do not end in zero under both iterations but instead end in loops are members of OEIS A219963:

A219963

Numbers n which do not reach zero under either of the iterations: \( n \rightarrow \lfloor \sqrt{n} \rfloor \times  (n - \lfloor \sqrt{n}) \rfloor\ ^2) \) or  \( n \rightarrow \lceil \sqrt{n} \rceil \times ( \lceil \sqrt{n} \rceil ^2- n ) \).

Not surprisingly, these numbers are rather rare. Here are the initial members of the sequence:

2897, 3159, 3183, 4004, 6335, 7025, 8163, 8237, 8621, 9234, 12204, 12963, 13381, 14340, 15217, 16191, 16438, 17474, 17763, 17972, 18065, 18990, 19677, 19848, 20345, 20803, 21426, 21539, 22022, 25834, 26872, 27175, 28052, 28929, 28996, 29295, 30511, 30991

The first pair of consecutive integers belonging to this sequence is (89971, 89972). As with all these sorts of trajectories, the issue of records arise. For example, what number produces the longest sequence? What number produces the maximum peak? The calculation at each step of the algorithm seems somewhat processor intensive so that calculations over large ranges cause it to time out (on SageMathCell at least). Still, these are interesting statistics to explore. It will be 303 days before my diurnal age again matches a member of this sequence.