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

Author

Jwalin Bhatt, Jul 06 2025

Keywords

• nonn

Comments

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!).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

Mathematica

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 *)

Python

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))