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 sequentially sampling the Gauss-Kuzmin distribution. - Jwalin Bhatt, Nov 28 2024

Links

Table of n, a(n) for n=1..104.

Prog

(Python)
import math
def sample_gauss_kuzmin_distribution(num_coeffs):
  coeffs, counts = [], [0]
  for _ in range(num_coeffs):
    min_time = math.inf
    for i, count in enumerate(counts, start=1):
      time = (count+1) / -math.log2(1-(1/((i+1)**2)))
      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.

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