Multiplicative partition

Multiplicative partition (Number theory)

In number theory, a multiplicative partition of an integer n that is greater than 1 is a way of writing n as a product of integers greater than 1. The number n is itself considered one of these products.

Examples

• 2*2*5, 2*10, 4*5, and 20 are the four multiplicative partitions of 20.
• 3*3*3*3, 3*3*9, 3*27, 9*9, and 81 are the five multiplicative permutations of 81 = 3^4. Because 81 is the fourth power of a prime, 81 has the same number (five) of multiplicative partitions as the number four has of (additive) partitions.
• 30 = 2*3*5 = 2*15 = 6*5 = 3*10 = 30 has five multiplicative partitions.

see also

partition (number theory)

divisor

References

• On the Number or Multiplicative Partitions, by John F. Hughes and J. O. Shallit. From The American Mathematical Monthly, August-September 1983.
• On a problem of Oppenheim concerning "factorisatio numerorum". By E. R. Canfield, Paul Erdos, Carl Pomerance. From Journal of Number Theory 17 (1983), no. 1 1--28.