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The fixed point of the iterations of the map x -> A093653(x) that start at n.
2

%I #11 May 24 2024 16:26:59

%S 1,2,3,3,3,6,3,3,3,6,3,3,3,3,3,3,3,6,3,3,3,3,3,3,6,3,3,3,3,6,6,6,3,6,

%T 3,3,3,3,6,3,3,6,3,3,3,6,6,3,3,3,3,3,3,6,3,3,6,6,6,3,6,3,3,3,3,3,3,3,

%U 6,6,3,3,3,3,3,3,3,3,6,3,3,3,3,3,3,6,3

%N The fixed point of the iterations of the map x -> A093653(x) that start at n.

%C Except for n = 1 and 2, all terms are either 3 or 6.

%C Do the asymptotic densities of the occurrences of 3 and 6 exist? The numbers of occurrences of 6 for n that do not exceed 10^k, for k = 1, 2, ..., are 2, 24, 234, 2735, 25321, 242398, 2605532, 27441386, 268518855, 2561508455, ... .

%H Amiram Eldar, <a href="/A373093/b373093.txt">Table of n, a(n) for n = 1..10000</a>

%e The iterations for the n = 1..7 are:

%e n a(n) iterations

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

%e 1 1 1

%e 2 2 2

%e 3 3 3

%e 4 3 4 -> 3

%e 5 3 5 -> 3

%e 6 6 6

%e 7 3 7 -> 4 -> 3

%t d[n_] := DivisorSum[n, Plus @@ IntegerDigits[#, 2] &]; a[n_] := FixedPointList[d, n][[-1]]; Array[a, 100]

%o (PARI) a(n) = {while(6 % n, n = sumdiv(n, d, hammingweight(d))); n;}

%Y Cf. A093653, A373092.

%K nonn,easy,base

%O 1,2

%A _Amiram Eldar_, May 23 2024