A385686 -id:A385686 - cp's OEIS frontend

A385685 Sequence where k is appended after every k! occurrences of 1, with multiple values following a 1 listed in order.

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

Offset: 0

Jwalin Bhatt, Jul 06 2025

nonn

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

The geometric mean approaches A385686 in the limit. In general, for parameter value p it approaches Product_{k>=2} k^(((p^(k-1))*((e-1)^(-p)))/k!).

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

Jwalin Bhatt, Table of n, a(n) for n = 0..9999
Wikipedia, Poisson distribution

Cf. A000142 (n!), A382093, A385686.

A385685[n_] := Module[{N1 = 0,NR= 1, result = {},i=1}, While[Length[result] < n,N1++;AppendTo[result, 1]; Do[If[Mod[N1, Factorial[k]] == 0,    AppendTo[result, k];f[k == NR + 1, NR++]],{k, 2, NR + 1}];If[Length[result] > n, result = Take[result, n]]];result];A385685[92] (* James C. McMahon, Jul 11 2025 *)

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) == 0:
                yield num
                num_reached += num == num_reached+1
A385685 = list(islice(poisson_distribution_generator(), 120))

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

A385685 Sequence where k is appended after every k! occurrences of 1, with multiple values following a 1 listed in order.

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