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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.

This sequence gives the number of iterations of A329623 for start value n before a repeated value first appears. Unlike the "ghost iteration" of A329200 the only fixed points are the single digits 0 to 9. See A328865 for the first repeating value.

Due to A329623(n) being significantly larger than n for some values of n, the iterative sequence can grow to infinity for some n. The first value to do so is n = 1373. This appears due to the occurrence of the digit string '62637' at the end of the term of the third iteration. These five digits reappear every second iteration at the end of the term, but with more and more digits preceding it. A329917 lists other divergent n values.

The smallest value, for n >= 10, which acts as an end point before repeating is 9, which is the final value for many starting values.

The digit string '8091' forms the basis of a very long convergent series for some values of n. The digit string consisting of an arbitrary number of copies of '80' followed by '91' will eventually converge to 8091, then 891, then 91, which finally converges in ten more iterations. We can thus form a number of arbitrary length using this rule which is guaranteed to converge. This sequence appears naturally with the starting value n = 139100 which converges to 9 after 136 iterations after reaching a term with 72 digits after 20 iterations. See the linked file below.

From M. F. Hasler, Dec 03 2019: (Start)

It seems the a(n) = -1 are conjectural, i.e., we have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates appear to explode up to a length of 157 bits, but finally end up in a 2-cycle of 41-digit numbers (26...26273, 62...62637).

The "repeating values" are members of cycles, listed in A328142. Only fixed points 1, ..., 9 and 4*(10^k-1)/9 + 11, k >=3, and 6 infinite families of 2-cycles are known.

(End)

The first escape value is a(1373) = -1 (without proof). - Georg Fischer, Jul 16 2020

Or: Number of iterations of A329200 until a number is seen for the second time in the trajectory of n.

A329200 consists of adding to n the number whose digits are the differences of adjacent digits of n in case it is even, or subtracting it if it is odd.

The trajectory of most small numbers ends in a repdigit (A010785) which are fixed points of this map. Some larger numbers enter nontrivial cycles, cf. examples and A329196. In both cases, some number(s) will appear infinitely often in the trajectory. This sequence gives the number of iterations until a value is repeated for the first time in the trajectory of n. This is also the size of n's orbit, i.e. the total number of distinct values that will ever appear.

If n is part of the cycle, a(n) gives the length of the cycle; in particular a(n) = 1 for fixed points.

For 11 <= n <= 99 the pattern (1, 2, 5, 2, 4, 2, 3, 2, 3, 2, 3) of length 11 repeats, i.e., a(n) = a(n') if n = n' (mod 11). But the trajectory of congruent n with same a(n) does not always end in the corresponding repdigit, e.g., 11+2 and 22+2 both end in 22, 33+2 ends in 33, 44+2 ends in 44, 55+2 and 66+2 both end in 66, 77+2 and 88+2 in 77, etc.

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.

A329623(n) = |n - A053392(n)|, where A053392 is the concatenation of the sums of pairs of consecutive digits.

This is the first number which appears twice for the iteration of A329623 starting with n. See A329624 for an explanation of the sequence, and for the number of iterations required before reaching this value. Terms a(9) to a(127) are all 9's, after which the sequence shows large jumps in value, e.g., a(1673) = 62626262626262626262626262637.

All -1 are so far conjectural, see A329624 and A329917 for more information.

The terms > 0 of this sequence are elements of cycles for A329623. Only 2-cycles and fixed points 1, 2, ..., 9 and 4...455 are known. Therefore a(n) is the earliest A329623^{k}(n) = A329623^{k+2}(n) if such k exist. See A328142 for the list of all possible values and more precise definitions. - M. F. Hasler, Dec 06 2019

The first escape value is a(1373) = -1. - Georg Fischer, Jul 16 2020

These are the starting values n for which the trajectory under iterations of A329623 grows without limit. See A329624 for an explanation of the sequence.

There are 466 terms below 10000.

From M. F. Hasler, Dec 02 2019: (Start)

There is no term below 10^3, but beyond 10^4 they become much more frequent: roughly 1/3 of all numbers between 10^4 and 5*10^4 are in the sequence.

The sequence consists mostly in runs of consecutive numbers ending with the next larger term with final digit 9: 1373..1379, 1382..1389, 1391..1399, 1591..1599, 1891..1899, 2373..2379, ... In some ranges, like a(152) = 4010 to a(240) = 4181, a(307) = 6010 to a(352) = 6391, a(467) = 10010 to a(556) = 11090, ..., this pattern is reversed: the runs start with a multiple of 10.

However, all these terms are so far only conjectural. We have no proof that the terms for which the trajectory seems to "explode" do not eventually end up in a cycle. For example, the 8th iterate of a(1) = 1373 is 5218725017016262626273. If the 2nd digit is changed from 2 to 0, then the further iterates grow to a length of 52 digits, but finally end up in a 2-cycle of 45-digit numbers (26...26273, 62...62637). (All members of cycles are listed in A328142.)

(End)

This sequence is based on the following iterative cycle. Start with n, set m = A040115(n), the concatenation of the absolute values of differences between adjacent digits, and then repeat the following until the number m has been previously seen: if m is greater than n, let m = A040115(m), otherwise let m = A053392(m), the concatenation of the sums of pairs of adjacent digits.

For all starting values n the iteration eventually converges to 0 or else goes into a cycle of finite length. When the number m gets larger than the iteration's starting value n it will always have its magnitude decreased by the operation m = A040115(m), while m = A053392(m) can either increase or decrease its magnitude, depending on the digit values of m. This has the overall effect of never allowing the iterative values to increase without limit as is seen in the similar iterations A328975 and A329624.

All the values of A053393 are seen as repeating values in this sequence, although this sequence has significantly more; probably an infinite number, although this is unknown. The first nonzero repeating value is not seen until a(9090), which forms the two-member loop of 999 -> 1818 -> 999. The first starting value that leads to an m value greater than the initial starting value is a(10090), see examples below. A330159 lists the starting values which are also the first repeating value.

For the first 20 million terms the longest iterative sequence is seen for a(18505180) which takes 457 steps before reaching 0. See attached link. The longest found looping sequence is for a(14106482) which reaches 1040103 after 5 steps and then again after 116 steps, forming a loop of length 111. The largest number found which starts the repeating loop is for a(9265011) which reaches 1411131715 after 9 iterations and then again after 41 iterations.

From a(12) to a(99) the sequence repeats a pattern of ten 3's followed by a 2. After that, a(100) = 4 and the terms begin to show a slow average increase in value.

Or: Number of iterations of A329201 until a number is seen for the second time in the trajectory of n.

A329201 consists of subtracting from or adding to n, depending on whether it is even or odd, the number A040115(n) whose digits are the differences of adjacent digits of n.

The trajectory of all numbers < 8000 ends in a repdigit (A010785), which are fixed points of this map. Some larger numbers enter nontrivial cycles, cf. A329342. In both cases, some number(s) will appear infinitely often in the trajectory. This sequence gives the number of iterations until a value is repeated for the first time in the trajectory of n. This is also the size of n's orbit, i.e. the total number of distinct values that will occur.

If n is part of a cycle (n in A329342), a(n) gives the length of the cycle; in particular a(n) = 1 for fixed points.

For 11 <= n <= 99 the pattern ( 1, 3, 2, 3, 2, 3, 2, 4, 2, 5, 2) of length 11 repeats. But the trajectory of those n with same a(n) does not always end in the corresponding repdigit.

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 sequences lists the self-repeating starting values n for the iterative sequence defined in A328680 up to starting values of n = 10^8. Each number in this sequence, when acting as the starting value for the A328680 iteration, will be the first number repeated in the iteration. Note that the other numbers appearing in the iterative sequence for a given start value n will in general NOT be other entries of this sequence as the iteration depends critically on the start value of n itself. As can be seen these numbers are quite rare, there being only 19 entries for n up to 100 million. It is unknown if this is a finite or infinite sequence.