A386016 -id:A386016 - cp's OEIS frontend

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

Paul D. Hanna, May 31 2003

nonn

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

			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 |     5.884     |       -       |       -       |   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)

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Gauss Kuzmin distribution

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

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 *)

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

A386009 Decimal expansion of Product_{k>=2} k^( ((k/2)^(k-1)) / (exp(k/2)*k!) ).

Original entry on oeis.org

1, 5, 4, 9, 2, 5, 1, 0, 4, 3, 4, 2, 5, 6, 3, 4, 0, 4, 8, 4, 0, 0, 5, 4, 4, 0, 2, 8, 1, 5, 8, 9, 4, 3, 5, 5, 5, 3, 0, 8, 8, 3, 5, 2, 6, 7, 0, 1, 5, 8, 3, 6, 8, 5, 4, 7, 2, 3, 3, 4, 6, 6, 4, 9, 5, 4, 7, 2, 2, 4, 6, 5, 2, 1, 8, 3, 3, 4, 7, 1, 6, 1, 9, 4, 2

Offset: 1

Jwalin Bhatt, Jul 14 2025

The geometric mean of the Borel distribution with parameter value 1/2 (A386016) approaches this constant. In general, for parameter value p it approaches Product_{k>=2} k^(((p*k)^(k-1))/((e^(p*k))*k!)).

			1.549251043425634048400544028158943555...

Cf. A386016.

Exp[NSum[((k/2)^(k-1) * Log[k])/(E^(k/2) * Factorial[k]), {k, 2, Infinity}, WorkingPrecision -> 120, NSumTerms -> 1000]]

Equals exp( Sum_{k>=2} log(k) * (((k/2)^(k-1)) / (exp(k/2)*k!)) ).

A386904 A sequence constructed by greedily sampling the Borel distribution for parameter value 1/2 to minimize discrepancy.

Original entry on oeis.org

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

Offset: 1

Jwalin Bhatt, Aug 07 2025

The geometric mean approaches A386009 in the limit.

The Borel distribution with parameter value 1/2 has PDF p(i) = (i/2)^(i-1) / (exp(i/2)*i!).

			Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far, then the expected occurrences of k at n-th step are given by n*p(k).
We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value.
| n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
|---|---------------|---------------|---------------|--------|
| 1 |     0.606     |       -       |       -       |   1    |
| 2 |     0.213     |     0.367     |       -       |   2    |
| 3 |     0.819     |    -0.448     |     0.251     |   1    |
| 4 |     0.426     |    -0.264     |     0.334     |   1    |
| 5 |     0.032     |    -0.080     |     0.418     |   3    |

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

Cf. A386009, A386016.

samplePDF[n_]:=Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
  coeffs=ConstantArray[0, n]; unreachedVal=1; counts=<||>;
  Do[probCountDiffs=Table[probCountDiff[i, k, counts], {i, 1, unreachedVal}];
    mostProbable=First@FirstPosition[probCountDiffs, Max[probCountDiffs]];
    If[mostProbable==unreachedVal, unreachedVal++]; coeffs[[k]]=mostProbable;
    counts[mostProbable]=Lookup[counts, mostProbable, 0]+1; , {k, 1, n}]; coeffs]
A386904=samplePDF[120]

cp's OEIS Frontend

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.

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A386009 Decimal expansion of Product_{k>=2} k^( ((k/2)^(k-1)) / (exp(k/2)*k!) ).

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A386904 A sequence constructed by greedily sampling the Borel distribution for parameter value 1/2 to minimize discrepancy.

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Keywords

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Crossrefs

Programs

Mathematica