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In mathematics, an integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once. For example, the sequence of powers of two , the basis of the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 1001012 = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include: * The even numbers; since adding even numbers produces only even numbers, no odd number can be formed. * Powers of three; no integer having a digit "2" in its ternary representation (2, 5, 6...) can be formed. == Conditions for completeness == Without loss of generality, assume the sequence ''a''''n'' is in nondecreasing order, and define the partial sums of ''a''''n'' as: :. Then the conditions : : are both necessary and sufficient for ''a''''n'' to be a complete sequence.〔Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985, pp.123-128.〕 A corollary to the above states that : : are sufficient for ''a''''n'' to be a complete sequence.〔 However there are complete sequences that do not satisfy this corollary, for example , consisting of the number 1 and the first prime after each power of 2. 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「Complete sequence」の詳細全文を読む スポンサード リンク
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