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===Factorizations===
If a number ''<math>N''</math> is a [[squarefree]] positive [[integer]], (meaning that it is the product of some number ''<math>n''</math> of distinct [[prime number]]s), then ''B<submath>nB_n</submath>'' gives the number of different [[multiplicative partition]]s of ''<math>N''</math>. These are [[factorization]]s of ''<math>N''</math> into numbers greater than one, treating two factorizations as the same if they have the same factors in a different order.<ref>{{harvnb|Williams|1945}} credits this observation to Silvio Minetola's ''Principii di Analisi Combinatoria'' (1909).</ref> For instance, 30 is the product of the three primes 2, 3, and 5, and has ''B''<submath>3B_3</submath> = 5 factorizations:
:<math>30 = 2\times 15=3\times 10=5\times 6=2\times 3\times 5</math>
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