559 ページ

フラン数第四形態改三は巨大数大好きbotが2018年3月22日に考案した東方巨大数である[1]。第2回東方巨大数に投稿された。近似値は$$f_{\varepsilon_4^{\varepsilon_4^{\varepsilon_4}}}(5)$$。

## 定義

• X, X' は任意の文字列
• Y は 0 個以上の ]
• n, A は自然数

1. o n = n
2. X-o n = X (n+1)
3. X[]Y n = XY n
4. X[X'-o]Y n = X[X']-o[X']-o ... -o[X']Yn ([X'] は n+1個)
5. X[X'<0>o]Y n = X[X'-o[-o[...[-o]]]]Y n (-o は n個)
6. X[X'<A>o]Y n = X[X'<A-1>o[<A-1>o[...[<A-1>o]]]]Y n (<A-1>はn個)

フラン数第四形態改三 = o-o[<5>o]5

## 近似

• o-o n = n+1
• o-o-o n = n+2
• o[-o]3 = o-o-o-o-o 3 = 7
• o[-o]n = 2n + 1
• o[-o-o]n = o[-o]-o[-o]-o...-o n ≈ 2^n
• o[-o-o-o]n = o[-o-o]-o[-o-o]-o...-o n ≈ 2↑↑n
• o[-o-o...-o]n = $$f_m(n+1)-2$$ (-oはm個) [3]
• o[-o[-o]]n = o[-o-o-o ... -o] n $$\approx f_\omega(n)$$
• o[-o[-o]]-o n $$\approx f_\omega(n+1)$$
• o[-o[-o]]-o[-o]n $$\approx f_\omega(2n)$$
• o[-o[-o]]-o[-o-o]n $$\approx f_\omega(2^n)$$
• o[-o[-o]]-o[-o[-o]]n $$\approx f_\omega^2(n)$$
• o[-o[-o]-o]2 = o[-o[-o]]-o[-o[-o]]2 $$\approx f_{\omega+1}(2)$$
• o[-o[-o]-o]n $$\approx f_{\omega+1}(n)$$
• o[-o[-o]-o-o]n $$\approx f_{\omega+2}(n)$$
• o[-o[-o]-o-o-o]n $$\approx f_{\omega+3}(n)$$
• o[-o[-o][-o]]n = o[-o[-o]-o-o ... -o]n $$\approx f_{\omega 2}(n)$$
• o[-o[-o][-o][-o]]n $$\approx f_{\omega 3}(n)$$
• o[-o[-o-o]]n = o[-o[-o]-o[-o] ... -o[-o]]n $$\approx f_{\omega^2}(n)$$
• o[-o[-o-o-o][-o-o][-o][-o]-o]n $$\approx f_{\omega^3 + \omega^2 + \omega 2 + 1}(n)$$
• o[-o[-o[-o]]]n = o[-o[-o-o ... -o]]n $$\approx f_{\omega^\omega}(n)$$
• o[-o[-o[-o]]]n $$\approx f_{\omega^\omega}(n)$$
• o[-o[-o[-o]]-o]n $$\approx f_{\omega^\omega+1}(n)$$
• o[-o[-o[-o]-o]]n $$\approx f_{\omega^{\omega+1}}(n)$$
• o[-o[-o[-o-o]]]n $$\approx f_{\omega^{\omega^2}}(n)$$
• o[-o[-o-[-o[-o]]]]n $$\approx f_{\omega^{\omega^\omega}}(n)$$
• o[-o[-o-[-o[-o[-o]]]]]n $$\approx f_{\omega^{\omega^{\omega^\omega}}}(n)$$
• o[<0>o]n = o[-o[-o[...[-o]]]]n $$\approx f_{\varepsilon_0}(n)$$
• o[<0>o-o]n $$\approx f_{\varepsilon_0+1}(n)$$
• o[<0>o-<0>o]n $$\approx f_{\varepsilon_0 2}(n)$$
• o[<0>o[-o]]n = o[<0>o-<0>o-<0>o ... -<0>o]n $$\approx f_{\varepsilon_0 \omega}(n)$$
• o[<0>o[-o[-o]]]n $$\approx f_{\varepsilon_0 \omega ^\omega}(n)$$
• o[<0>o[<0>o]]n $$\approx f_{\varepsilon_0 ^2}(n)$$
• o[<0>o[<0>o][<0>o]]n $$\approx f_{\varepsilon_0 ^3}(n)$$
• o[<0>o[<0>o-o]n $$\approx f_{\varepsilon_0 ^\omega}(n)$$
• o[<0>o[<0>o-o[-o]]n $$\approx f_{\varepsilon_0 ^{\omega^\omega}}(n)$$
• o[<0>o[<0>o[<0>o]]]n $$\approx f_{\varepsilon_0 ^{\varepsilon_0}}(n)$$
• o[<0>o[<0>o[<0>o]][<0>o]]n $$\approx f_{\varepsilon_0 ^{\varepsilon_0+1}}(n)$$
• o[<0>o[<0>o[<0>o][<0>o]]]n $$\approx f_{\varepsilon_0 ^{\varepsilon_0 2}}(n)$$
• o[<0>o[<0>o[<0>o[<0>o]]]]n $$\approx f_{\varepsilon_0 ^{\varepsilon_0^2}}(n)$$
• o[<0>o[<0>o[<0>o[<0>o[<0>o]]]]]n $$\approx f_{\varepsilon_0 ^{\varepsilon_0^{\varepsilon_0}}}(n)$$
• o[<1>o]n = o[<0>o[<0>o ... [<0>o]]]n $$\approx f_{\varepsilon_1}(n)$$
• o[<2>o]n $$\approx f_{\varepsilon_2}(n)$$
• o[<3>o]n $$\approx f_{\varepsilon_3}(n)$$
• o[<4>o]n $$\approx f_{\varepsilon_4}(n)$$
• o[<5>o]5 = o[<4>o[<4>o[<4>o[<4>o[<4>o]]]]]5 $$\approx f_{\varepsilon_4 ^{\varepsilon_4^{\varepsilon_4}}}(5)$$