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单词 SumproductNumber
释义

sum-product number


A sum-product number is an integer n that in a given base satisfies the equality

n=i=1mdii=1mdi

where di is a digit of n, and m is the number of digits of n. This means a test of whether the sum of the digits of n times the product of the digits of n is equal to n.

For example, the number 128 in base 7 is a sum-product number since

2427=(2+4+2)(242)

All sum-product numbers are Harshad numbers, too.

0 and 1 are sum-product numbers in any positional base. The proof that the set of sum-product numbers in base 2 is finite is elementary enough not to inspire claims of authorship. The proof that the set of sum-product numbers in base 10 is finite (specifically, 0, 1, 135 and 144) is more involved but within the realm of basic algebra, and it points the way to a formulation of the proof that number of sum-product numbers in any base is finite.

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更新时间:2025/5/4 21:58:03