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

1,5

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.

Links

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

Wikipedia, Zeta distribution

Example

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

Prog

(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))
(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

Crossrefs

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

Sequence in context: A029434 A358192 A396405 * A156281 A002217 A344173

Adjacent sequences: A381519 A381520 A381521 * A381523 A381524 A381525

Keyword

nonn

Author

Jwalin Bhatt, Feb 26 2025

Status

approved