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In number theory, Sylvester's sequence is an integer sequence in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are: :2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 . Sylvester's sequence is named after James Joseph Sylvester, who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series of unit fractions that converges to 1 more rapidly than any other series of unit fractions with the same number of terms. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. Values derived from this sequence have also been used to construct finite Egyptian fraction representations of 1, Sasakian Einstein manifolds, and hard instances for online algorithms. == Formal definitions == Formally, Sylvester's sequence can be defined by the formula : The product of an empty set is 1, so ''s''0 = 2. Alternatively, one may define the sequence by the recurrence : with ''s''0 = 2. It is straightforward to show by induction that this is equivalent to the other definition. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sylvester's sequence」の詳細全文を読む スポンサード リンク
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