This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A084580 #40 Jul 30 2025 00:31:50
%S A084580 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,
%T A084580 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,
%U A084580 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
%N 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.
%C A084580 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,...].
%C A084580 This can also be defined as the sequence formed by greedily sampling the Gauss-Kuzmin distribution. - _Jwalin Bhatt_, Nov 28 2024
%H A084580 Jwalin Bhatt, <a href="/A084580/b084580.txt">Table of n, a(n) for n = 1..10000</a>
%H A084580 Wikipedia, <a href="https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribution">Gauss Kuzmin distribution</a>
%e A084580 From _Jwalin Bhatt_, Jul 25 2025: (Start)
%e A084580 Constructing the sequence by greedily sampling the Gauss-Kuzmin distribution to minimize discrepancy.
%e A084580 Let p(n) denote the probability of n and c(n) denote the count of occurrences of n so far.
%e A084580 We take the ratio of the actual occurrences c(n)+1 to the probability and pick the one with the lowest value.
%e A084580 Since p(n) is monotonic decreasing, we only need to compute c(n) once we see c(n-1).
%e A084580 | n | (c(1)+1)/p(1) | (c(2)+1)/p(2) | (c(3)+1)/p(3) | choice |
%e A084580 |---|---------------|---------------|---------------|--------|
%e A084580 | 1 | 5.884 | - | - | 1 |
%e A084580 | 2 | 4.818 | 5.884 | - | 1 |
%e A084580 | 3 | 7.228 | 5.884 | - | 2 |
%e A084580 | 4 | 7.228 | 11.769 | 10.740 | 1 |
%e A084580 | 5 | 9.637 | 11.769 | 10.740 | 1 |
%e A084580 | 6 | 12.047 | 11.769 | 10.740 | 3 | (End)
%t A084580 pdf[k_] := Log[1 + 1/(k*(k + 2))]/Log[2]
%t A084580 samplePDF[numCoeffs_] := Module[
%t A084580 {coeffs = {}, counts = {0}, minTime, minIndex, time},
%t A084580 Do[
%t A084580 minTime = Infinity;
%t A084580 Do[
%t A084580 time = (counts[[i]] + 1)/pdf[i];
%t A084580 If[time < minTime, minIndex = i; minTime = time],
%t A084580 {i, 1, Length[counts]}
%t A084580 ];
%t A084580 If[minIndex == Length[counts], AppendTo[counts, 0]];
%t A084580 counts[[minIndex]] += 1;
%t A084580 AppendTo[coeffs, minIndex],
%t A084580 {numCoeffs}
%t A084580 ];
%t A084580 coeffs
%t A084580 ]
%t A084580 A084580 = samplePDF[120] (* _Jwalin Bhatt_, Jul 25 2025 *)
%o A084580 (Python)
%o A084580 import math
%o A084580 def sample_gauss_kuzmin_distribution(num_coeffs):
%o A084580 coeffs, counts = [], [0]
%o A084580 for _ in range(num_coeffs):
%o A084580 min_time = math.inf
%o A084580 for i, count in enumerate(counts, start=1):
%o A084580 time = (count+1) / -math.log2(1-(1/((i+1)**2)))
%o A084580 if time < min_time:
%o A084580 min_index, min_time = i, time
%o A084580 if min_index == len(counts):
%o A084580 counts.append(0)
%o A084580 counts[min_index-1] += 1
%o A084580 coeffs.append(min_index)
%o A084580 return coeffs
%o A084580 A084580 = sample_gauss_kuzmin_distribution(100) # _Jwalin Bhatt_, Dec 22 2024
%Y A084580 Cf. A002210, A084576-A084579, A084581-A084587, A372869, A386016.
%K A084580 nonn
%O A084580 1,3
%A A084580 _Paul D. Hanna_, May 31 2003