A393793 A sequence constructed by greedily sampling the discrete probability distribution p(k)=zeta(k+1)-1 to minimize discrepancy.

1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 5, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 6, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 2, 1, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 5, 1, 2, 1, 1, 1, 2

Offset

1,2

Comments

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

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.

Example

Let p(k) denote the probability of k and c(k) denote the number of occurrences of k among the first n-1 terms; then the expected number of occurrences of k among n random terms is 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 |     0.645     |       -       |       -       |   1    |
| 2 |     0.289     |     0.404     |       -       |   2    |
| 3 |     0.934     |    -0.393     |     0.246     |   1    |
| 4 |     0.579     |    -0.191     |     0.329     |   1    |
| 5 |     0.224     |     0.010     |     0.411     |   3    |

Mathematica

probCountDiff[j_, k_, count_] := k*(Zeta[j+1] -1) - Lookup[count, j, 0]
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]
A393793 = samplePDF[120]

Prog

(Python)
from sympy import zeta
def prob_count_diff(j, k, count):
    return  k*(zeta(j+1)-1) - count
def sample_zeta_distribution(num_coeffs):
    coeffs, unreached_val, counts = [], 1, {}
    for k in range(1, num_coeffs+1):
        prob_count_diffs = [prob_count_diff(i, k, counts.get(i, 0)) for i in range(1, unreached_val+1)]
        most_probable = prob_count_diffs.index(max(prob_count_diffs)) + 1
        unreached_val += most_probable == unreached_val
        coeffs.append(most_probable)
        counts[most_probable] = counts.get(most_probable, 0) + 1
    return coeffs
A393793 = sample_zeta_distribution(120)

Crossrefs

Cf. A393663.

Sequence in context: A394352 A137457 A396505 * A275699 A305633 A214123

Adjacent sequences: A393790 A393791 A393792 * A393794 A393795 A393796

Keyword

nonn

Author

Jwalin Bhatt, Feb 27 2026

Status

approved