A352540 Values for which the iteration of A352544 (half if even, add largest anagram if odd) does not end in a loop.

89, 109, 117, 137, 149, 178, 187, 203, 205, 207, 209, 213, 217, 218, 223, 225, 234, 239, 247, 253, 255, 257, 267, 273, 274, 277, 279, 293, 295, 297, 298, 299, 307, 319, 327, 335, 347, 356, 365, 374, 405, 406, 407, 409, 410, 414, 415, 418, 426, 427, 434, 436, 437, 445, 446

Offset: 1

Author

M. F. Hasler, Mar 20 2022

Keywords

• nonn
• base

Comments

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.

Examples

			See A352541 for examples of trajectories which end in a loop, and A352542 for the trajectory of 89 which grows to infinity.
		

Links

• Eric Angelini, Divide by 2 or add the biggest anagram, math-fun discussion list, Mar 20 2022.

Crossrefs

Cf. A352544 (iterated map: half if even, add largest anagram if odd), A352541 (number of iterations to see a value again), A352542 (trajectory of 89), A352543 (starting values ending in cycles of length > 2), A352545 (representatives of cycles of length > 2).

Programs

PARI

select( {is_A352540(n)=!A352541(n)}, [0..500])

Formula

{ n >= 0 | A352541(n) = 0 }.