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A352544 is a variant of the Collatz map, where for an odd argument x, the number A004186(x) (= digits of x arranged in decreasing order) is added.

The first zero appears for initial value n = 89. See A352542 for the trajectory of n = 89. See A352540 for the indices of zeros.

89 is the smallest nonnegative integer with an orbit of infinite size under iterations of x -> A352544(x) = {x/2 if x is even, x + A004186(x) if x is odd}. The list of all such numbers is given in A352540, which contains this sequence as subset.

We conjecture that there is a strictly increasing sequence (b(k), k >= 0) = (32, 37, 46, 52, 88, 91, 118, 122, 141, ...) such that all terms a(n) with n >= b(k) have more than k digits 0.

As a consequence, the sequence tends to a 10-adic limit ...27057751007.

Similarly, the number of leading digits 1 appears to grow to infinity; more precisely, a(n) has more than k leading digits 1 for all n > c(k >= 0) = (50, 70, 95, 121, 122, 123, 130, ...).

The iterated map A352544 is a variant of the Collatz map, A352544(x) = x/2 if x is even, A352544(x) = x + A004186(x) (add x with digits in decreasing order) if x is odd.

All the terms are only conjectured to have this property; we don't have a completely rigorous proof. But for all the listed initial terms, the trajectory quickly reaches numbers with many (>> 10) digits and grows larger with every iteration: When the number is odd and has a digit 0, then its successor is again odd and at least twice as large, most often more than 9 times larger. Roughly 1/10th of the digits are zeros, and similarly for 9s, so as the terms get larger, it becomes increasingly less probable that they could end up having no digit 0 at all, which is only a necessary condition that they might become even and not grow upon for one iteration, but still most likely resume growth immediately after. See sequence A352542, the trajectory of a(1) = 89, for an example studied in detail.

Most small starting values end in a cycle or loop of length 2 under iterations of A352544 (like 1 -> 2 -> 1, 3 -> 6 -> 3, or 49 -> 143 -> ... -> 7915 -> 17666 -> 8833 -> 17666), and some (listed in A352540) have an unbounded orbit like 89 (cf. A352542).

This sequence lists all other starting values, i.e., those which have a bounded orbit but don't end in a cycle of length 2. (A352544 obviously has no fixed points.)

All terms below a(54) = 9203 lead to the same loop 1611 -> 7722 -> 3861 -> 12492 -> 6246 -> 3123 -> 6444 -> 3222 -> 1611 of size 8.

The starting value 9203 is the only one below 10^4 leading to a different loop, of size 22: cf. EXAMPLE.

Sequence A352545 lists the representatives (smallest elements) of the distinct cycles of length > 2.

All terms are odd, since the smallest (resp. largest) element of a cycle of A352544 is always odd (resp. even).

3886083 is also in the sequence, cf. EXAMPLE.

a(5) > 500000.

To compute the sequence we look for odd initial values ending in a cycle of more than two elements. This gives terms of the sequence, but we don't know the position of a term until we have scanned all (relevant) initial values up to that number, cf. EXAMPLES.