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

1,2

Comments

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)!).

This sequence can be transformed to A385685 by subtracting 1 from every term and then deleting every occurrence of 0. So [1,2,1,2,3] -> [1,1,2]. - Jwalin Bhatt, Mar 18 2026

This sequence is generated by the D'Hondt (or Jefferson) apportionment method for the Poisson distribution translated by 1 unit (choosing the smallest element in case of ties). - Pontus von Brömssen, Mar 30 2026

Links

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

Wikipedia, D'Hondt method.

Wikipedia, Poisson distribution.

Example

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.

Prog

(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-1) == 0:
                yield num
                num_reached += num == num_reached+1
A382093 = list(islice(poisson_distribution_generator(), 120))

Crossrefs

Cf. A381522, A381900, A382095, A385685.

Sequence in context: A325530 A171712 A091412 * A106036 A007001 A094917

Adjacent sequences: A382090 A382091 A382092 * A382094 A382095 A382096

Keyword

nonn

Author

Jwalin Bhatt, Mar 25 2025

Status

approved