OFFSET
1,9
COMMENTS
The frequencies of the terms follow the zeta distribution with parameter value 3. That is, the asymptotic frequency of the integer k equals 1/(zeta(3)*k^3).
The geometric mean approaches exp(-zeta'(3)/zeta(3)) = 1.1791840098089555... in the limit. In general, if the sequence was formed by every k^s occurrences, it would approach e^(-zeta'(s)/zeta(s)).
Unlike A381522, this sequence has a finite arithmetic mean, equal to zeta(2)/zeta(3) = 1.368432777... and in general for power s it equals zeta(s-1)/zeta(s).
Properties change sharply at s=2, the geometric mean remains finite for all s but the arithmetic mean becomes finite for s > 2 and infinite for s <= 2.
| mean | 1 < s <= 2 | s > 2 |
| ---------- | ---------- | ------ |
| arithmetic | oo | finite |
| geometric | finite | finite |
Considered as an irregular triangle, the n-th row lists the divisors of the cube root of the largest cube dividing n.
This sequence is generated by the D'Hondt (or Jefferson) apportionment method for the given distribution.
LINKS
Jwalin Bhatt, Table of n, a(n) for n = 1..10000
Wikipedia, D'Hondt method.
Wikipedia, Zeta distribution.
EXAMPLE
After every 8 ones we see a 2, after every 27 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 ** 3) == 0:
yield num
num_reached += num == num_reached + 1
A397556 = list(islice(zeta_distribution_generator(), 87))
(PARI) lista(nn)={my(L=List()); for(n=1, nn, my(t=1, f=factor(n)); for(i=1, #f~, t*=f[i, 1]^(f[i, 2]\3)); fordiv(t, d, listput(L, d))); Vec(L[1..nn])}
CROSSREFS
KEYWORD
nonn
AUTHOR
Jwalin Bhatt, Jun 30 2026
STATUS
approved