A381900 Sequence where k is appended after every (2^(k-1))*k occurrences of 1, with multiple values following a 1 listed in order.

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

Offset: 1

Author

Jwalin Bhatt, Mar 09 2025

Keywords

• nonn

Comments

The frequencies of the terms follow the logarithmic distribution with parameter value 1/2.

The geometric mean approaches A381898 in the limit.

Examples

			After every (2*2=4) ones we see a 2,
after every (4*3=12) ones we see a 3,
after every (8*4=32) ones we see a 4 and so on.
		

Links

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

Crossrefs

Cf. A381522, A381898.

Programs

Python

from itertools import islice
def logarithmic_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 % ((2**(num-1))*(num)) == 0:
                yield num
                num_reached += num == num_reached+1
A381900 = list(islice(logarithmic_distribution_generator(), 120))