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Sequence A040115 is most naturally extended to 0 (empty sum) for single-digit arguments; that's what we use here. This value is added to n if even, subtracted if odd.

Repdigit numbers are the fixed points. Other starting values end in nontrivial loops under iterations of this map, like 11090 -> 10891 -> 12709 -> 11130 -> 11107 -> 11090 etc. Table A329196 lists these cycles, A329197 their lengths.

A329198 gives the size of n's orbit, i.e., the length of the trajectory until the terminating cycle is covered.

A329200 consists in adding or subtracting the number A040115(n) whose digits are the differences of adjacent digits of n, depending on its parity.

Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence gives the length of these cycles, ordered by their smallest member, as they are listed in the table A329196. See there for more information.

A329200 consists of adding the number whose digits are the absoute values of differences of adjacent digits of n in case it is even, or subtracting it if it is odd. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. This sequence lists these cycles, ordered by their smallest member which is always listed first. Sequence A329197 gives the row lengths.

Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the (cyclic) first differences are also the same and add again up to 0.) This is the case for rows 1, 2, 3, ... (but not row 4 or 6) of this table. Rows 5, 7 and 8 are the second members of these three families. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.

A329201 consists of adding or subtracting the number whose digits are the differences of adjacent digits of n, depending on its parity. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles.

This sequence lists these cycles, ordered by their smallest member which is always listed first.

Sequence A329341 gives the lengths of these cycles, i.e., rows of this table.

Whenever all terms of a cycle have the same number of digits and same initial digit, then this digit can be prefixed k times to each term to obtain a different cycle of same length, for any k >= 0. (The corresponding "ghosts" A040115(n) are then the same, so the first differences are also the same and add again up to 0.) This is the case for rows 3, 4, 5, 6, ... of this table. Rows 7, 8, 11, ... are subsequent members of the respective family. We could call "primitive" the cycles which are not obtained from an earlier cycle by duplicating the initial digits.

A329201 consists of adding or subtracting the number A040115(n) whose digits are the difference between adjacent digits of n, depending on its parity. Repdigits A010785 are fixed points of this map, but some numbers enter nontrivial cycles. Sequence A329342 lists these cycles, ordered by their smallest member which is always listed first. This sequence gives the row lengths.

This sequence lists the number of iterations required of the sequence A330633, the concatenation of the product of adjacent digits, for starting value n until the resulting term equals zero. If the iterative cycle never reaches zero and instead diverges with larger and larger numbers with each iteration then a(n) = -1.

For larger values of n most starting values lead to a divergent series. The first such term is a(166). See the examples below. This reaches 888 after three iterations which is guaranteed to diverge due to the iterative cycle 888 -> 6464 -> 242424 -> 88888, and the series of 8's expands forever. All starting values examined up to 10^8 either reach zero else eventually contain three or more adjacent 8's and will therefore diverge. Unlike similar iterative sequences A329624 and A329198 there appears to be no fixed points or other cyclic behavior.

Up to n = 10^8 the largest number of iterative steps before reaching zero is 23. This is first seen for n = 3178 and for 18520 other starting values. Considering no larger number of steps was found, nor any during a brief test of random numbers larger than 10^8, it is possible this is the largest number of steps to reach zero for all starting values, with the exception of the repunits comprising k 1's which will take k steps to reach zero. A random number with a large number of digits will almost certainly diverge as it will result in 888, or similar divergent numbers, appearing within the iterative term at some step which, as shown above, will diverge.

There are numbers other than repunits of k 1's which take k steps to reach 0, e.g. n = 691...16 with m 1's where m is odd or 491...16 where m is even has a(n) = m + 17. - Robert Israel, Jul 01 2024