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Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$, where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing $\odot$ with other operators.

$\odot = +$ just results in $(n+1)n/2$. E.g., for $n=10$, the plus-factorial is $55$.

$\odot = -$ results in $-n(n-1)/2$. E.g., for $n=10$, the minus-factorial is $-35$. Here (and below) I am associating to the left, i.e., $((4-3)-2)-1=-2$.

So I explored the binary operation $ a \odot b = \sqrt{a b}$, a geometric-mean operation. Then the geometric-mean-factorial $gm!(n)$ looks like this: $$gm!(2) = \sqrt{2 \cdot 1} \approx 1.41421$$ $$gm!(3) = \sqrt{(\sqrt{3 \cdot 2}) \cdot 1} = 6^{\frac{1}{4}} \approx 1.56508$$ $$gm!(4) = \sqrt{(\sqrt{(\sqrt{4 \cdot 3}) \cdot 2}) \cdot 1}=\sqrt{2} \cdot 3^{\frac{1}{8}} \approx 1.62239$$ $$\ldots$$ $$gm!(20) \approx 1.66169$$ The geometric-mean-factorial seems to be approaching a limit that is unfamiliar to me. Does anyone recognize this constant from another context?

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The references provided by guest will likely have explored the following suggestion: Evaluate log of your constant as approximately .5078, or the limit of sum ln(i)/2^i . Gerhard "Adding Seems Simpler Than Multiplying" Paseman, 2013.12.30 –  Gerhard Paseman Dec 31 '13 at 1:28
In fact, the difference between $\sum_{1 \leq i \leq k} \log(i)/2^i$ and $\log(gm!(k+1))$ is sufficiently small that you might find the sum easier to handle. Gerhard "These Things Take A While" Paseman, 2013.12.30 –  Gerhard Paseman Dec 31 '13 at 1:59
Actually the MathWorld link also suggests what the generalization of this function to real (or complex) arguments might be, e.g., by trying to solve $g(z+1)=zg(z)^2$ on wolframalpha.com –  Suvrit Dec 31 '13 at 3:11
Those generalized factorials can be interpreted as the length of paths in graphs; the integers would resemble the labels of vertices and the binary operation yields the weight of directed edges. My question related to shortest paths in the "Cantor Graph" originated in such an interpretation. One could now ask for shortest paths in the arithmetic-geometric mean graph and maybe obtain further interesting results. –  Manfred Weis Jan 1 at 12:26
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up vote 13 down vote accepted

looks like Somos's quadratic recurrence constant

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Nice find---Thanks! There is a nice article in MathWorld at this link, which also details Gerhard's log-approach. –  Joseph O'Rourke Dec 31 '13 at 2:10
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