This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352540 #17 Mar 30 2022 06:17:00
%S A352540 89,109,117,137,149,178,187,203,205,207,209,213,217,218,223,225,234,
%T A352540 239,247,253,255,257,267,273,274,277,279,293,295,297,298,299,307,319,
%U A352540 327,335,347,356,365,374,405,406,407,409,410,414,415,418,426,427,434,436,437,445,446
%N A352540 Values for which the iteration of A352544 (half if even, add largest anagram if odd) does not end in a loop.
%C A352540 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.
%C A352540 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.
%H A352540 Eric Angelini, <a href="https://mailman.xmission.com/hyperkitty/list/math-fun@mailman.xmission.com/thread/RI5P2MYI7NKD32LK47XTYTGUAEMQKVVZ/">Divide by 2 or add the biggest anagram</a>, math-fun discussion list, Mar 20 2022.
%F A352540 { n >= 0 | A352541(n) = 0 }.
%e A352540 See A352541 for examples of trajectories which end in a loop, and A352542 for the trajectory of 89 which grows to infinity.
%o A352540 (PARI) select( {is_A352540(n)=!A352541(n)}, [0..500])
%Y A352540 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).
%K A352540 nonn,base
%O A352540 1,1
%A A352540 _M. F. Hasler_, Mar 20 2022