A084580 Let y = m/GK(k), where m and k vary over the positive integers and GK(k)=log(1+1/(k*(k+2)))/log(2) is the Gauss-Kuzmin distribution of k. Sort the y values by size and number them consecutively by n. This sequence gives the values of k in order by n.

1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 4, 2, 1, 3, 1, 2, 1, 5, 1, 1, 2, 1, 3, 6, 1, 4, 2, 1, 1, 1, 2, 3, 1, 7, 1, 2, 1, 5, 1, 4, 2, 1, 3, 1, 8, 1, 2, 1, 1, 3, 2, 1, 6, 1, 4, 9, 1, 2, 1, 5, 1, 3, 2, 1, 1, 1, 2, 10, 1, 4, 3, 1, 7, 2, 1, 1, 1, 2, 1, 3, 5, 1, 11, 2, 6, 1, 4, 1, 2, 1, 3, 1, 1, 8, 2, 1, 1, 12, 2, 1, 3, 4, 1, 1

Offset

1,3

Comments

The geometric mean of the sequence equals Khintchine's constant K=2.685452001 = A002210 since the frequency of the integers agrees with the Gauss-Kuzmin distribution. When considered as a continued fraction, the resulting constant is 0.5815803358828329856145... = A372869 = [0;1,1,2,1,1,3,2,1,1,1,4,2,1,...].

This can also be defined as the sequence formed by greedily sampling the Gauss-Kuzmin distribution. - Jwalin Bhatt, Nov 28 2024

The sequence is also generated by the D'Hondt (or Jefferson) apportionment method for the given distribution. - Pontus von Brömssen, Mar 30 2026

Links

Jwalin Bhatt, Table of n, a(n) for n = 1..10000

Wikipedia, D'Hondt method.

Wikipedia, Gauss Kuzmin distribution.

Example

From Jwalin Bhatt, Jul 25 2025: (Start)
Constructing the sequence by greedily sampling the Gauss-Kuzmin distribution to minimize discrepancy.
Let p(n) denote the probability of n and c(n) denote the count of occurrences of n so far.
We take the ratio of the actual occurrences c(n)+1 to the probability and pick the one with the lowest value.
Since p(n) is monotonic decreasing, we only need to compute c(n) once we see c(n-1).
  | n | (c(1)+1)/p(1) | (c(2)+1)/p(2) | (c(3)+1)/p(3) | choice |
  |---|---------------|---------------|---------------|--------|
  | 1 |     2.409     |       -       |       -       |   1    |
  | 2 |     4.818     |     5.884     |       -       |   1    |
  | 3 |     7.228     |     5.884     |       -       |   2    |
  | 4 |     7.228     |    11.769     |    10.740     |   1    |
  | 5 |     9.637     |    11.769     |    10.740     |   1    |
  | 6 |    12.047     |    11.769     |    10.740     |   3    | (End)

Mathematica

pdf[k_] := Log[1 + 1/(k*(k + 2))]/Log[2]
samplePDF[numCoeffs_] := Module[
  {coeffs = {}, counts = {0}, minTime, minIndex, time},
Do[
    minTime = Infinity;
    Do[
      time = (counts[[i]] + 1)/pdf[i];
      If[time < minTime, minIndex = i; minTime = time],
      {i, 1, Length[counts]}
    ];
    If[minIndex == Length[counts], AppendTo[counts, 0]];
    counts[[minIndex]] += 1;
    AppendTo[coeffs, minIndex],
    {numCoeffs}
  ];
  coeffs
]
A084580 = samplePDF[120]  (* Jwalin Bhatt, Jul 25 2025 *)

Prog

(Python)
from mpmath import iv
log2 = iv.log(2)
def sample_gauss_kuzmin_distribution(num_coeffs):
  coeffs, counts = [], [0]
  for _ in range(num_coeffs):
    min_time = iv.inf
    for i, count in enumerate(counts, start=1):
      time = (count+1) / -(iv.log(1-(1/((i+1)*(i+1))))/log2)
      if time < min_time:
        min_index, min_time = i, time
    if min_index == len(counts):
      counts.append(0)
    counts[min_index-1] += 1
    coeffs.append(min_index)
  return coeffs
A084580 = sample_gauss_kuzmin_distribution(100) # Jwalin Bhatt, Dec 22 2024

Crossrefs

Cf. A002210, A084576-A084579, A084581-A084587, A372869, A386016.

Sequence in context: A030305 A228350 A220482 * A263633 A171850 A356144

Adjacent sequences: A084577 A084578 A084579 * A084581 A084582 A084583

Keyword

nonn

Author

Paul D. Hanna, May 31 2003

Status

approved