A381522 Sequence where k is appended after every k^2 occurrences of 1, with multiple values following a 1 listed in order.

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

Offset: 1

Author

Jwalin Bhatt, Feb 26 2025

Keywords

• nonn

Comments

The frequencies of the terms follow the zeta distribution with parameter value 2.

The geometric mean approaches exp(-zeta'(2)/zeta(2)) A381456 in the limit. In general, if the sequence was formed by every k^s occurrences, it would approach e^(-zeta'(s)/zeta(s)).

Considered as an irregular triangle, the n-th row lists the divisors of the square root of the largest square dividing n.

Examples

			After every 4 ones we see a 2, after every 9 ones we see a 3 and so on.
		

Links

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

Crossrefs

Cf. A000188, A084580, A381456, A381900, A382093.

Programs

PARI

lista(n)={my(L=List()); for(n=1, n, fordiv(sqrtint(n/core(n)), d, listput(L,d))); Vec(L[1..n])} \\ Andrew Howroyd, Feb 26 2025

Python

from itertools import islice
def zeta_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) == 0:
                yield num
                num_reached += num == num_reached+1
A381522 = list(islice(zeta_distribution_generator(), 120))