A393663 A sequence constructed by greedily sampling the discrete probability distribution p(k) = zeta(k+1)-1 to minimize ratio discrepancy.

1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 2, 1, 1, 5, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 1

Offset

1,4

Comments

The geometric mean approaches Product_{k>=2} k^(zeta(k+1)-1) = 1.40665264499... in the limit.

This sequence is generated by the D'Hondt (or Jefferson) apportionment method for the given distribution. - Pontus von Brömssen, Mar 31 2026

Links

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

William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Wikipedia, D'Hondt method.

Example

Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far.
We take the ratio of the actual occurrences c(k)+1 to the probability and pick the one with the lowest value.
Since p(k) is monotonic decreasing, we only need to compute c(k) once we see c(k-1).
| n | (c(1)+1)/p(1) | (c(2)+1)/p(2) | (c(3)+1)/p(3) | choice |
|---|---------------|---------------|---------------|--------|
| 1 |     1.550     |       -       |       -       |   1    |
| 2 |     3.101     |     4.949     |       -       |   1    |
| 3 |     4.651     |     4.949     |       -       |   1    |
| 4 |     6.202     |     4.949     |       -       |   2    |
| 5 |     6.202     |     9.898     |    12.147     |   1    |

Mathematica

pdf[i_] := Zeta[i+1] -1
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
]
A393663 = samplePDF[120]

Prog

(Python)
from sympy import oo, zeta
def sample_zeta_distribution(num_coeffs):
  coeffs, counts = [], [0]
  for _ in range(num_coeffs):
    min_time = oo
    for i, count in enumerate(counts, start=1):
      time = (count+1) / ((zeta(i+1))-1)
      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
A393663 = sample_zeta_distribution(120)

Crossrefs

Cf. A381617, A386016.

Sequence in context: A182937 A340035 A386016 * A185147 A206921 A386261

Adjacent sequences: A393660 A393661 A393662 * A393664 A393665 A393666

Keyword

nonn

Author

Jwalin Bhatt, Feb 24 2026

Status

approved