A382095 -id:A382095 - cp's OEIS frontend

A382093 Sequence where k is appended after every (k-1)! occurrences of 1, with multiple values following a 1 listed in order.

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

Offset: 1

Jwalin Bhatt, Mar 25 2025

nonn

The frequencies of the terms follow the Poisson distribution with parameter value 1.

The geometric mean approaches A382095 in the limit. In general, for parameter value p it approaches Product_{k>=2} k^(((p^(k-1))*(e^(-p)))/(k-1)!).

			Every 1 is followed by a 2 because (2-1)! = 1,
after every (2!=2) ones we see a 3,
after every (3!=6) ones we see a 4 and so on.

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

Cf. A381522, A381900, A382095.

from itertools import islice
from math import factorial
def poisson_distribution_generator():
    num_ones, num_reached = 0, 1
    while num_ones := num_ones+1:
        yield 1
        for num in range(2, num_reached+2):
            if num_ones % factorial(num-1) == 0:
                yield num
                num_reached += num == num_reached+1
A382093 = list(islice(poisson_distribution_generator(), 120))

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.

Original entry on oeis.org

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

Jwalin Bhatt, Apr 20 2025

nonn

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

			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    |

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

Cf. A381617, A382093, A382095, A382961.

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]

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A382093 Sequence where k is appended after every (k-1)! occurrences of 1, with multiple values following a 1 listed in order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula