Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied by Paul Erdős and others.[1][2][3]
The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers. This means that any factorial can be written as a product of powers of primes. For example,
If we wish to write
as a product of factors of the form
, where each
is a prime number, and the factors are sorted in nondecreasing order, then we have three ways of doing so:
The number of such "sorted multiplicative partitions" of
grows with
, and is given by the sequence
- 1, 1, 3, 3, 10, 10, 30, 75, 220, 220, 588, 588, 1568, 3696, 11616, ... (sequence A085288 in the OEIS).
Not all sorted multiplicative partitions of a given factorial have the same length. For example, the partitions of have lengths 4, 3 and 5. In other words, exactly one of the partitions of has length 5. The number of sorted multiplicative partitions of that have length equal to is 1 for and , and thereafter increases as
- 2, 2, 5, 12, 31, 31, 78, 78, 191, 418, 1220, 1220, 3015, ... (sequence A085289 in the OEIS).
Consider all sorted multiplicative partitions of that have length , and find the partition whose first factor is the largest. (Since the first factor in a partition is the smallest within that partition, this means finding the
maximum of all the minima
.) Call this factor
. The value of
is 2 for
and
, and thereafter grows as
- 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, ... (sequence A085290 in the OEIS).
To express the asymptotic behavior of , let
As
tends to infinity,
approaches a limiting value, the
Alladi–Grinstead constant (named for the mathematicians
Krishnaswami Alladi and Charles Grinstead). The
decimal representation of the Alladi–Grinstead constant begins,
0.80939402054063913071793188059409131721595399242500030424202871504... (sequence A085291 in the OEIS).
The exact value of the constant can be written as the exponential of a certain infinite series. Explicitly,[4]
where
is given by
This sum can alternatively be expressed as follows,
[5] writing
for the
Riemann zeta function:
This series for the constant
converges more rapidly than the one before.
[5] The function
is constant over stretches of
, but jumps from 5 to 7, skipping the value 6. Erdős raised the question of how large the gaps in the sequence of
can grow, and how long the constant stretches can be.
[3][6]
References