A373093 The fixed point of the iterations of the map x -> A093653(x) that start at n.

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, 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, 6, 6, 3, 3, 3, 3, 3, 3, 3, 3, 6, 3, 3, 3, 3, 3, 3, 6, 3

Offset: 1

Author

Amiram Eldar, May 23 2024

Keywords

• nonn
• easy
• base

Comments

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

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, ... .

Examples

			The iterations for the n = 1..7 are:
  n  a(n)  iterations
  -  ----  -----------
  1    1   1
  2    2   2
  3    3   3
  4    3   4 -> 3
  5    3   5 -> 3
  6    6   6
  7    3   7 -> 4 -> 3
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A093653, A373092.

Programs

Mathematica

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

PARI

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