A383855 -id:A383855 - cp's OEIS frontend

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

2, 2, 0, 0, 1, 6, 1, 0, 5, 8, 0, 9, 9, 0, 2, 6, 5, 5, 3, 1, 9, 4, 5, 5, 7, 8, 6, 6, 5, 5, 9, 9, 4, 4, 8, 7, 2, 6, 8, 5, 6, 6, 2, 3, 2, 4, 7, 5, 2, 7, 2, 3, 8, 8, 8, 7, 2, 3, 1, 4, 5, 1, 1, 7, 7, 6, 3, 1, 6, 9, 0, 1, 1, 2, 6, 9, 6, 6, 5, 9, 4, 7, 5, 8, 4, 7, 0, 2, 9, 7, 3, 4, 7, 2, 6, 8, 0, 7, 6, 2, 5, 8, 1, 6, 1

Offset: 1

Jean-François Alcover, Jul 15 2014

The geometric mean of the Yule-Simon distribution with parameter value 1 (A383855) approaches this constant. In general, the geometric mean of the Yule-Simon distribution approaches Product_{k>=2} k^(1/(p*Beta(k,p+1))). - Jwalin Bhatt, May 12 2025

			2.200161058099026553194557866559944872685662324752723888723145117763169...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8.1 Alternative representations [of real numbers], p. 62.

Sofia Kalpazidou, Khintchine's constant for Lüroth representation, Journal of Number Theory, Vol. 29, No. 2 (June 1988), pp. 196-205.

Cf. A002210, A085291, A085361, A242624, A244109(V), A131688(W), A383855.

evalf(exp(Sum((Zeta(n+1)-1)/n, n=1..infinity)), 120); # Vaclav Kotesovec, Dec 11 2015

Exp[NSum[Log[k]/(k*(k+1)), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 5000, Method -> {NIntegrate, MaxRecursion -> 100}]] (* Vaclav Kotesovec, Dec 11 2015 *)

Equals exp(A085361).
U*V*W = 1, where V is A244109 and W is A131688.
Equals e * A085291. - Amiram Eldar, Jun 27 2021
Equals 1/A242624. - Amiram Eldar, Feb 06 2022

Corrected by Vaclav Kotesovec, Dec 11 2015

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.

Original entry on oeis.org

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, 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, 1, 2, 1, 5, 1, 15, 1, 2, 1, 7, 1, 3, 1, 2, 1, 16, 1, 17

Offset: 1

Jwalin Bhatt, May 14 2025

nonn

The geometric mean approaches A245254 in the limit.

The probability mass function of the Yule-Simon distribution with parameter 1 is given by p(k) = 1/(k*(k+1)) for k >= 1.

			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 |     0.5       |     0.166     |     0.083     |   1    |
| 2 |     0         |     0.333     |     0.166     |   2    |
| 3 |     0.5       |    -0.5       |     0.25      |   1    |
| 4 |     0         |    -0.333     |     0.333     |   3    |
| 5 |     0.5       |    -0.166     |    -0.583     |   1    |

Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, Yule-Simon distribution

Cf. A245254, A383855.

probCountDiff[j_, k_, count_]:=k/(j*(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]
A383899=samplePDF[120]

cp's OEIS Frontend

A245254 Decimal expansion of U = Product_{k>=1} (k^(1/(k*(k+1)))), a Khintchine-like limiting constant related to Lüroth's representation of real numbers.

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