A381617 A sequence constructed by greedily sampling the zeta distribution for parameter value 2 to minimize discrepancy.

1, 2, 1, 1, 3, 1, 4, 1, 1, 2, 1, 1, 5, 1, 2, 1, 1, 6, 1, 3, 1, 1, 2, 1, 7, 1, 1, 8, 1, 2, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 9, 1, 2, 1, 1, 10, 1, 2, 1, 1, 3, 1, 5, 1, 1, 2, 1, 11, 1, 1, 4, 1, 2, 1, 1, 3, 1, 1, 2, 1, 12, 1, 1, 13, 1, 2, 1, 1, 3, 1, 6, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 14

Offset: 1

Author

Jwalin Bhatt, Mar 10 2025

Keywords

• nonn

Comments

The geometric mean approaches exp(-zeta'(2)/zeta(2)) = A381456 in the limit.

The zeta distribution PDF is p(i) = 1/(zeta(2)*i^2).

Examples

			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 |     0.607     |       -       |       -       |   1    |
| 2 |     0.215     |     0.303     |       -       |   2    |
| 3 |     0.823     |    -0.544     |     0.202     |   1    |
| 4 |     0.431     |    -0.392     |     0.270     |   1    |
| 5 |     0.039     |    -0.240     |     0.337     |   3    |
		

Links

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

Crossrefs

Cf. A241773, A381522.

Programs

Mathematica

probCountDiff[j_, k_, count_] := k/(Zeta[2] j^2) - 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]
A381617 = samplePDF[120]