A385685 -id:A385685 - cp's OEIS frontend

A385686 Decimal expansion of exp((Sum_{k>=2} log(k)/k!)/(e-1)).

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

Offset: 1

Jwalin Bhatt, Jul 06 2025

The geometric mean of the Poisson distribution with parameter value 1 (A385685) approaches this constant.

			1.4210379597319607153378144890592856953982...

Cf. A296301, A306243, A382095, A385685.

N[Exp [Sum[Log[i]/Factorial[i], {i, 2, Infinity}] / (E-1) ], 120]

PARI
```
prodinf(k=2, k^(1/k!))^(1/(exp(1)-1))
```

Equals exp((Sum_{k>=2} log(k)/k!)/(e-1)).
Equals (Product_{k>=2} k^(1/k!)) ^ (1/(e-1)).
From Vaclav Kotesovec, Jul 08 2025: (Start)
Equals exp(A306243/(exp(1) - 1)).
Equals A296301^(1/(exp(1) - 1)). (End)

A385873 A sequence constructed by greedily sampling the Poisson distribution for parameter value 1 so as to minimize discrepancy.

Original entry on oeis.org

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

Offset: 1

Jwalin Bhatt, Jul 11 2025

nonn

The geometric mean approaches A385686 = exp((Sum_{k>=2} log(k)/k!)/(e-1)) in the limit.

The Poisson distribution used here is p(i) = 1/((e-1)*i!).

			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 |     0.581     |     0.290     |     0.096     |   1    |
| 2 |     0.163     |     0.581     |     0.193     |   2    |
| 3 |     0.745     |    -0.127     |     0.290     |   1    |
| 4 |     0.327     |     0.163     |     0.387     |   3    |
| 5 |     0.909     |     0.454     |    -0.515     |   1    |

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

Cf. A383238, A385685, A385686.

probCountDiff[j_, k_, count_]:=N[k/((E-1)*Factorial[j])]-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]
A385873=samplePDF[120]

cp's OEIS Frontend

A385686 Decimal expansion of exp((Sum_{k>=2} log(k)/k!)/(e-1)).

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