A factorial square product is a product of distinct factorials that is a square
number. For an integer 
, let 
be the least 
for which there are integers 
such that
 |
(1)
|
for an integer 
.
Set 
if there is no representation. For example, 
since 
,
and 
since 
. No prime number
has a representation, while every composite number
has 
(Erdős and Graham 1976).
Define the counting function for real 
by
 |
(2)
|
Problem 374 in the Erdős problems collection asks for the orders of growth of these functions for 
(Bloom 2026). Yudin (2026) proved, for every 
,
that
 |
(3)
|
where 
denotes big-O notation and
 |
(4)
|
with 
and 
the squarefree part of 
. Each distinct member of 
is counted once. The classical bound for 
(Erdős and Graham 1976, p. 346) and Yudin's (2026)
bounds for 
and 
give
 |
(5)
|
Here 
means bounded above and below by positive constant
multiples for large 
.
Together with 
and 
for 
, this determines all orders of growth. The notation
denotes the floor function. Whether 
has a limit for 
, 5, and 6 remains open (Yudin 2026).
The integers requiring six factors begin 527, 611, 713, 731, 779, 893, 923, 1003, 1037, ... (OEIS A389148).
