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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

Given a prime p, the class of sequences a(n,p) can be constructed from linear combination of the two sequences b(n,p) (A010885) and c(n,p) (A226233), according to a(n,p) = c(n,p)*p - b(n,p) (see Formula below) that ensures uniqueness of the form q = a(n,p)*p^m according to the decomposition theorem Vaseghi 2013 (see link and reference below), for p prime, q a positive integer and m a positive integer or zero. The above example is for p=7. The class is crucial and will be applied to define other number theoretic sequences, that will be submitted to OEIS as well a posterior.