A383238 A sequence constructed by greedily sampling the Poisson distribution for parameter value 1, 1/(e*(i-1)!) to minimize discrepancy selecting the smallest value in case of ties.

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

Offset: 1

Author

Jwalin Bhatt, Apr 20 2025

Keywords

• nonn

Comments

The geometric mean approaches A382095 = exp(Sum_{k>=2} log(k)/(k-1)!) in the limit.

Examples

			Let p(k) denote the probability of k and c(k) denote the count of occurrences of k so far, then the expected occurrences of k at n-th step are 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 |     0.367     |     0.367     |     0.183     |   1    |
| 2 |    -0.264     |     0.735     |     0.367     |   2    |
| 3 |     0.103     |     0.103     |     0.551     |   3    |
| 4 |     0.471     |     0.471     |    -0.264     |   1    |
| 5 |    -0.160     |     0.839     |    -0.080     |   2    |
		

Links

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

Crossrefs

Cf. A381617, A382093, A382095, A382961.

Programs

Mathematica

probCountDiff[j_,k_,count_]:=N[k/(E*Factorial[j-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]
A383238=samplePDF[120]