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A383899 A sequence constructed by greedily sampling the Yule-Simon distribution for parameter value 1, to minimize discrepancy selecting the smallest value in case of ties.

%I A383899 #19 Jul 29 2025 00:18:04
%S A383899 1,2,1,3,1,4,1,2,1,5,1,6,1,2,1,3,1,7,1,2,1,8,1,9,1,2,1,4,1,3,1,2,1,10,
%T A383899 1,11,1,2,1,3,1,5,1,2,1,4,1,12,1,2,1,3,1,13,1,2,1,6,1,14,1,2,1,3,1,4,
%U A383899 1,2,1,5,1,15,1,2,1,7,1,3,1,2,1,16,1,17
%N A383899 A sequence constructed by greedily sampling the Yule-Simon distribution for parameter value 1, to minimize discrepancy selecting the smallest value in case of ties.
%C A383899 The geometric mean approaches A245254 in the limit.
%C A383899 The probability mass function of the Yule-Simon distribution with parameter 1 is given by p(k) = 1/(k*(k+1)) for k >= 1.
%H A383899 Jwalin Bhatt, <a href="/A383899/b383899.txt">Table of n, a(n) for n = 1..10000</a>
%H A383899 Wikipedia, <a href="https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution">Yule-Simon distribution</a>
%e A383899 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).
%e A383899 We subtract the actual occurrences c(k) from the expected occurrences and pick the one with the highest value.
%e A383899 | n | n*p(1) - c(1) | n*p(2) - c(2) | n*p(3) - c(3) | choice |
%e A383899 |---|---------------|---------------|---------------|--------|
%e A383899 | 1 |     0.5       |     0.166     |     0.083     |   1    |
%e A383899 | 2 |     0         |     0.333     |     0.166     |   2    |
%e A383899 | 3 |     0.5       |    -0.5       |     0.25      |   1    |
%e A383899 | 4 |     0         |    -0.333     |     0.333     |   3    |
%e A383899 | 5 |     0.5       |    -0.166     |    -0.583     |   1    |
%t A383899 probCountDiff[j_, k_, count_]:=k/(j*(j+1))-Lookup[count, j, 0]
%t A383899 samplePDF[n_]:=Module[{coeffs, unreachedVal, counts, k, probCountDiffs, mostProbable},
%t A383899   coeffs=ConstantArray[0, n]; unreachedVal=1; counts=<||>;
%t A383899   Do[probCountDiffs=Table[probCountDiff[i, k, counts], {i, 1, unreachedVal}];
%t A383899     mostProbable=First@FirstPosition[probCountDiffs, Max[probCountDiffs]];
%t A383899     If[mostProbable==unreachedVal, unreachedVal++]; coeffs[[k]]=mostProbable;
%t A383899     counts[mostProbable]=Lookup[counts, mostProbable, 0]+1; , {k, 1, n}]; coeffs]
%t A383899 A383899=samplePDF[120]
%Y A383899 Cf. A245254, A383855.
%K A383899 nonn
%O A383899 1,2
%A A383899 _Jwalin Bhatt_, May 14 2025