\(\uparrow\) More Up-Arrows \(\uparrow\)

Given two natural numbers \(a\) and \(b\), the following definitions hold:

  • \( a \uparrow\uparrow\uparrow b = \underbrace{a\uparrow\uparrow (a \uparrow\uparrow (\cdots \uparrow\uparrow a))}_{b\text{ times}}\).
  • \(a \uparrow\uparrow b = \underbrace{a\uparrow(a \uparrow (\cdots \uparrow a))}_{b\text{ times}}\).
  • \(a\uparrow b = a^b\).

Compute \( \displaystyle \sum_{n=1}^{9} \left(n\uparrow\uparrow\uparrow (n+1) \mod 10\right)\).

The following are some examples of these tetration function:

  • \(2\uparrow\uparrow 2 = 2^{2} = 4\).
  • \(3\uparrow\uparrow 3 = 3^{3^3} = 3^{27} = 7625597484987\).
  • \(4\uparrow\uparrow 4 =4^{4^{4^4}} = 4^{1.3408 \times 10^{154}} \approx 10^{10^{153.9}}\).
  • \(5\uparrow\uparrow 5 = 5^{5^{5^{5^5}}} \approx 10^{10^{10^{2184.1}}}\)
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