Multiplicative partition: Difference between revisions

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==Examples==

*The number 20 has four multiplicative partitions: 2 × 2 × 5, 2 × 10, 4 × 5, and 20.

*3 × 3 × 3 × 3, 3 × 3 × 9, 3 × 27, 9 × 9, and 81 are the five multiplicative permutations of 81 = 3⁴. Because 81 is the fourth power of a [[prime number|prime]], 81 has the same number (five) of multiplicative partitions as the number four has of (additive) [[partition (number theory)|partitions]].

*The number 30 has five multiplicative partitions: 2 × 3 × 5 = 2 × 15 = 6 × 5 = 3 × 10 = 30. In general, the number of multiplicatice partitions of a number with '''i''' distinct prime factors is the ith [[Bell number]] , B_i.

==Application==

{{harvtxt|Hughes|Shallit|1983}} describe an application of multiplicative partitions in classifying integers with a given number of divisors. For example, the integers with exactly 12 divisors take the forms ''p''¹¹, ''p''×''q''⁵, ''p''²×''q''³, and ''p''×''q''×''r''², where ''p'', ''q'', and ''r'' are distinct [[prime number]]s; these forms correspond to the multiplicative partitions 12, 2×6, 3×4, and 2×2×3 respectively. More generally, for each multiplicative partition