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.
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,formula_1If we wish to write formula_2 as a product of factors of the form formula_3, where each formula_4 is a prime number, and the factors are sorted in nondecreasing order, then we have three ways of doing so:formula_5The number of such "sorted multiplicative partitions" of formula_6 grows with formula_7, and is given by the sequence
1, 1, 3, 3, 10, 10, 30, 75, 220, 220, 588, 588, 1568, 3696, 11616, ... (sequence in the OEIS).
Not all sorted multiplicative partitions of a given factorial have the same length. For example, the partitions of formula_2 have lengths 4, 3 and 5. In other words, exactly one of the partitions of formula_2 has length 5. The number of sorted multiplicative partitions of formula_6 that have length equal to formula_7 is 1 for formula_12 and formula_13, and thereafter increases as
2, 2, 5, 12, 31, 31, 78, 78, 191, 418, 1220, 1220, 3015, ... (sequence in the OEIS).
Consider all sorted multiplicative partitions of formula_6 that have length formula_7, 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 formula_16. The value of formula_16 is 2 for formula_12 and formula_13, 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 in the OEIS).
To express the asymptotic behavior of formula_16, letformula_21As formula_7 tends to infinity, formula_23 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 in the OEIS).The exact value of the constant can be written as the exponential of a certain infinite series. Explicitly,formula_24where formula_25 is given byformula_26This sum can alternatively be expressed as follows, writing formula_27 for the Riemann zeta function:formula_28This series for the constant formula_25 converges more rapidly than the one before. The function formula_16 is constant over stretches of formula_7, but jumps from 5 to 7, skipping the value 6. Erdős raised the question of how large the gaps in the sequence of formula_16 can grow, and how long the constant stretches can be.