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Amenable Number

A number n is called amenable if it can be built up from integers a_1, a_2, ..., a_k by either addition or multiplication such that

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

(Tamvakis 1995).

The solutions are the numbers n such that n=0 or 1 (mod 4), excluding n=4 (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

Amenable, Composite Number, Partition, Sum

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References

Lossers, O. P. "Solution to Problem 10454. Amenable Numbers." Amer. Math. Monthly 105, 1998.Sloane, N. J. A. Sequence A100832 in "The On-Line Encyclopedia of Integer Sequences."Tamvakis, H. "Problem 10454." Amer. Math. Monthly 102, 463, 1995.

Referenced on Wolfram|Alpha

Amenable Number

Cite this as:

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

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