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Amenable Number

An amenable number is a positive integer n for which there are exactly n integers a_1, ..., a_n whose sum and product both equal n,

sum_(i=1)^na_i=product_(i=1)^na_i=n
(1)

(Tamvakis 1995). Repeated and negative integers are allowed.

The solutions are the numbers n such that n=0 or 1 (mod 4), excluding n=4 (Tamvakis and Lossers 1998), giving 1, 5, 8, 9, 12, 13, 16, 17, ... (OEIS A100832). For example, 5 and 8 are amenable since

5=1-1+1-1+5
(2)
=1×(-1)×1×(-1)×5
(3)
8=1-1+1-1+1+1+2+4
(4)
=1×(-1)×1×(-1)×1×1×2×4.
(5)

See also

Composite Number, Product, Sum

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References

Sloane, N. J. A. Sequence A100832 in "The On-Line Encyclopedia of Integer Sequences."Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102, 463, 1995. https://doi.org/10.2307/2975042.Tamvakis, H. and Lossers, O. P. "Amenable Numbers: 10454." Amer. Math. Monthly 105, 368, 1998. https://doi.org/10.2307/2589724.

Referenced on Wolfram|Alpha

Amenable Number

Cite this as:

Weisstein, Eric W. "Amenable Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AmenableNumber.html

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