A084576 -id:A084576 - cp's OEIS frontend

A084587 Integer floor of the reciprocal of Gauss-Kuzmin distribution of n: a(n) = floor( log(2)/log(1 + 1/(n*(n+2))) ).

2, 5, 10, 16, 24, 33, 44, 55, 68, 83, 99, 116, 135, 155, 177, 199, 224, 249, 276, 305, 335, 366, 398, 432, 468, 504, 543, 582, 623, 665, 709, 754, 800, 848, 897, 948, 1000, 1053, 1108, 1164, 1222, 1281, 1341, 1403, 1466, 1530, 1596, 1663, 1732, 1802, 1873, 1946

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084586.

for(n=1,100,a=floor(log(2)/log(1+1/(n*(n+2)))); print1(a,","))

a(n) = (n^2 + 2n)*log(2) + O(1). - Charles R Greathouse IV, Sep 04 2015

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.

Original entry on oeis.org

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

A084582 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 n when k=2.

Original entry on oeis.org

3, 7, 12, 16, 21, 27, 31, 36, 41, 47, 51, 58, 63, 67, 74, 78, 84, 89, 95, 99, 106, 110, 116, 123, 127, 133, 137, 142, 149, 156, 160, 166, 170, 177, 183, 187, 192, 199, 205, 210, 216, 221, 227, 232, 236, 243, 249, 256, 259, 265, 270, 278, 284, 287, 294, 297, 305

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084581, A084582-A084587.

A084577 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 floor(y) in order by n.

Original entry on oeis.org

2, 4, 5, 7, 9, 10, 11, 12, 14, 16, 16, 17, 19, 21, 21, 23, 24, 24, 26, 28, 29, 31, 32, 33, 33, 33, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 50, 52, 53, 53, 55, 55, 57, 58, 60, 62, 64, 64, 65, 67, 67, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 84, 85

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Missing positive integers are found in A084578.

Cf. A084576, A084578-A084587.

A084578 Positive integers not found in A084577.

Original entry on oeis.org

1, 3, 6, 8, 13, 15, 18, 20, 22, 25, 27, 30, 34, 37, 39, 46, 51, 54, 56, 59, 61, 63, 66, 71, 78, 80, 87, 90, 92, 95, 97, 102, 104, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 140, 143, 145, 148, 157, 160, 162, 165, 174, 179, 181, 184, 189, 191, 208, 210, 213

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576, A084577, A084579-A084587.

A084579 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 m in order by n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 2, 5, 6, 7, 1, 3, 8, 2, 9, 4, 10, 1, 11, 12, 5, 13, 3, 1, 14, 2, 6, 15, 16, 17, 7, 4, 18, 1, 19, 8, 20, 2, 21, 3, 9, 22, 5, 23, 1, 24, 10, 25, 26, 6, 11, 27, 2, 28, 4, 1, 29, 12, 30, 3, 31, 7, 13, 32, 33, 34, 14, 1, 35, 5, 8, 36, 2, 15, 37, 38, 39, 16, 40, 9, 4, 41, 1, 17, 3

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084578, A084580-A084587.

A084581 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 n when k=1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 15, 17, 19, 20, 22, 25, 28, 29, 30, 33, 35, 37, 39, 42, 44, 46, 48, 49, 52, 54, 57, 59, 61, 64, 65, 66, 69, 72, 75, 76, 77, 79, 82, 86, 88, 90, 92, 93, 96, 97, 100, 103, 104, 107, 108, 111, 113, 115, 119, 122, 124, 125, 126, 129, 131, 134, 136

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084580, A084582-A084587.

A084584 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 n when k=4.

Original entry on oeis.org

11, 26, 40, 55, 70, 87, 102, 118, 132, 148, 164, 180, 195, 213, 228, 245, 260, 276, 292, 308, 326, 342, 355, 375, 391, 407, 421, 439, 454, 473, 487, 505, 521, 536, 552, 568, 587, 602, 615, 636, 650, 669, 684, 699, 718, 732, 750, 767, 783, 802, 816, 831, 849

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084583, A084585-A084587.

A084585 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 n when k=5.

Original entry on oeis.org

18, 38, 60, 81, 105, 128, 152, 172, 196, 219, 242, 266, 289, 313, 337, 358, 385, 408, 431, 453, 479, 500, 525, 549, 574, 597, 617, 644, 671, 691, 716, 738, 762, 789, 811, 834, 858, 884, 908, 929, 956, 977, 1002, 1023, 1052, 1075, 1096

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084584, A084586-A084587.

A084586 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 n when k=6.

Original entry on oeis.org

24, 53, 85, 114, 146, 178, 209, 240, 273, 306, 338, 370, 401, 434, 467, 498, 531, 563, 596, 629, 660, 692, 724, 759, 793, 822, 856, 889, 921, 955, 987, 1018, 1056, 1085

Offset: 1

Paul D. Hanna, May 31 2003

nonn

Cf. A084576-A084585, A084587.

cp's OEIS Frontend

A084587 Integer floor of the reciprocal of Gauss-Kuzmin distribution of n: a(n) = floor( log(2)/log(1 + 1/(n*(n+2))) ).

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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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A084582 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 n when k=2.

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A084577 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 floor(y) in order by n.

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A084578 Positive integers not found in A084577.

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A084579 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 m in order by n.

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A084581 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 n when k=1.

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A084584 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 n when k=4.

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A084585 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 n when k=5.

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A084586 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 n when k=6.

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