A383855 The n-th term of the sequence is k after every k*(k+1)/2 occurrences of 1, with multiple values following a 1 listed in order.

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 1, 1, 1, 2, 5, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 6, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 7, 1, 1, 2, 3, 4, 5, 1, 1, 1, 2, 1, 1, 1, 2, 3, 8, 1, 1, 1, 2, 1, 4, 1, 1, 2, 3, 6, 1, 1, 1, 2, 5, 9, 1, 1, 1, 2, 3, 1, 1, 4, 1, 2, 1, 1, 1, 2, 3, 1, 10

Offset: 1

Author

Jwalin Bhatt, May 12 2025

Keywords

• nonn

Comments

The frequencies of the terms follow the Yule-Simon distribution with parameter value 1. The geometric mean approaches A245254 in the limit.

Examples

			After every ((2*3)/2=3) ones we see a 2,
after every ((3*4)/2=6) ones we see a 3,
after every ((4*5)/2=10) ones we see a 4 and so on.
		

Links

• Jwalin Bhatt, Table of n, a(n) for n = 1..10000
• Wikipedia, Yule-Simon distribution

Crossrefs

Cf. A245254, A383899.

Programs

Python

from itertools import islice
def beta_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 % (num*(num+1)//2) == 0:
                yield num
                num_reached += num == num_reached+1
A383855 = list(islice(beta_distribution_generator(), 120))