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
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
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
To express the asymptotic behavior of , letAs 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 byThis 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]
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
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
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
To express the asymptotic behavior of , letAs 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 byThis 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]