A373092 -id:A373092 - cp's OEIS frontend

A373094 a(n) is the least number k such that A373092(k) = n.

1, 4, 7, 12, 24, 120, 1260, 1829520

Offset: 0

Amiram Eldar, May 23 2024

a(n) is the least number k such that the number of iterations of the map x -> A093653(x) required to reach from k to a fixed point is n.

a(8) > 4*10^10.

			The iterations for the n = 0..7 are:
  n     a(n)  iterations
  -  -------  --------------------------------------------------
  0        1   1
  1        4   4 -> 3
  2        7   7 -> 4 -> 3
  3       12   12 -> 9 -> 5 ->3
  4       24   24 -> 12 -> 9 -> 5 -> 3
  5      120   120 -> 36 -> 15 -> 9 -> 5 -> 3
  6     1260   1260 -> 120 -> 36 -> 15 -> 9 -> 5 -> 3
  7  1829520   1829520 -> 1260 -> 120 -> 36 -> 15 -> 9 -> 5 -> 3

Cf. A093653, A095347 (decimal analog), A373092.

d[n_] := d[n] = DivisorSum[n, Plus @@ IntegerDigits[#, 2] &];
f[n_] := -2 + Length@ FixedPointList[d, n];
seq[len_] := Module[{s = Table[0, {len}], c = 0, i, n = 1}, While[c < len, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[7]

f(n) = {my(c = 0); while(6 % n, n = sumdiv(n, d, hammingweight(d)); c++); c;}
lista(len) = {my(s = vector(len), c = 0, i, n = 1); while(c < len, i = f(n) + 1; if(i <= len && s[i] == 0, c++; s[i] = n); n++); s;}

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

Original entry on oeis.org

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

Amiram Eldar, May 23 2024

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, ... .

			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

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

Cf. A093653, A373092.

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

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

cp's OEIS Frontend

A373094 a(n) is the least number k such that A373092(k) = n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI