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a(n) is the greatest value of the orbit of n under repeated application of A350229 (the sum of a number and its balanced ternary digits).
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%I #7 Jan 24 2022 16:12:09

%S 0,2,2,6,6,6,6,8,8,14,14,14,14,16,14,15,16,17,18,20,20,24,24,24,24,26,

%T 26,32,32,32,32,44,32,44,44,44,44,44,44,44,44,44,44,44,44,45,46,47,48,

%U 50,50,51,52,53,54,56,56,60,60,60,60,62,62,68,68,68,68

%N a(n) is the greatest value of the orbit of n under repeated application of A350229 (the sum of a number and its balanced ternary digits).

%C The sequence { a(n) - n, n >= 0 } has no upper limit (this because the sequence A065363 can be positive on arbitrarily large intervals).

%e For n = 9:

%e - the orbit of 9 contains the following values:

%e k v bter(v) ds(v)

%e - -- ------- -----

%e 0 9 100 1

%e 1 10 101 2

%e 2 12 110 2

%e 3 14 1TTT -2

%e 4 12 110 2

%e - so a(9) = max({ 9, 10, 12, 14 }) = 14.

%t f[n_] := n + Total[If[First@ # == 0, Rest@ #, #] &[Prepend[IntegerDigits[n, 3], 0] //. {x___, y_, k_ /; k > 1, z___} :> {x, y + 1, k - 3, z}]]; Array[Max@ NestWhileList[f, #, UnsameQ, All] &, 67, 0] (* _Michael De Vlieger_, Jan 15 2022 *)

%o (PARI) b(n) = my (v=n, d); while (n, n=(n-d=[0,1,-1][1+n%3])/3; v+=d); v

%o a(n) = my (s=[]); while (!setsearch(s, n), s=setunion(s, [n]); n=b(n)); s[#s]

%Y Cf. A350229, A350656.

%K nonn,base

%O 0,2

%A _Rémy Sigrist_, Jan 10 2022