# 雙重指數

## 雙重指數數列

${\displaystyle 2^{2^{n}}}$
${\displaystyle F_{m}=2^{2^{m}}+1}$
${\displaystyle M_{M_{p}}=2^{2^{p}-1}-1}$

Aho和Sloane發現有許多整數數列的每一項是前一項的平方再加上一個整數，這類的數列常常可以用最接近雙重指數數列的整數來表示，且雙重指數數列中間的指數為2[1]。若一整數數列的第n項和n的雙重指數成正比，Ionascu 及Stanica將這樣的整數數列稱為「幾乎雙重指數」（almost doubly-exponential），可以定義為雙重指數加上一常數後再取整數[2]

${\displaystyle s_{n}=\left\lfloor E^{2^{n+1}}+{\frac {1}{2}}\right\rfloor }$

${\displaystyle a(n)=\left\lfloor A^{3^{n}}\right\rfloor }$

## 應用

### 數論

${\displaystyle 2^{4^{n}}}$

## 參考資料

1. ^ Aho, A. V.; Sloane, N. J. A., Some doubly exponential sequences, Fibonacci Quarterly, 1973, 11: 429–437
2. ^ Ionascu, E.; Stanica, P., Effective asymptotics for some nonlinear recurrences and almost doubly-exponential sequences, Acta Mathematica Universitatis Comenianae, 2004, LXXIII (1): 75–87.
3. ^ Gruber, Hermann; Holzer, Markus. Finite Automata, Digraph Connectivity, and Regular Expression Size (PDF). Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2008): 39–50. 2008. doi:10.1007/978-3-540-70583-3_4.
4. ^ Nielsen, Pace P., An upper bound for odd perfect numbers, INTEGERS: the Electronic Journal of Combinatorial Number Theory, 2003, 3: A14.
5. ^ Miller, J. C. P.; Wheeler, D. J., Large prime numbers, Nature, 1951, 168 (4280): 838, Bibcode:1951Natur.168..838M, doi:10.1038/168838b0.