Definition:Euclid Number/Sequence

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Definition

The sequence of Euclid numbers begins as follows:

\(\displaystyle E_0 \ \ \) \(\displaystyle = p_0\# + 1\) \(=\) \(\displaystyle 1 + 1\) \(\displaystyle = 2\)
\(\displaystyle E_1 \ \ \) \(\displaystyle = p_1\# + 1\) \(=\) \(\displaystyle 2 + 1\) \(\displaystyle = 3\)
\(\displaystyle E_2 \ \ \) \(\displaystyle = p_2\# + 1\) \(=\) \(\displaystyle 2 \times 3 + 1\) \(\displaystyle = 7\)
\(\displaystyle E_3 \ \ \) \(\displaystyle = p_3\# + 1\) \(=\) \(\displaystyle 2 \times 3 \times 5 + 1\) \(\displaystyle = 31\)
\(\displaystyle E_4 \ \ \) \(\displaystyle = p_4\# + 1\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 + 1\) \(\displaystyle = 211\)
\(\displaystyle E_5 \ \ \) \(\displaystyle = p_5\# + 1\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11 + 1\) \(\displaystyle = 2311\)
\(\displaystyle E_6 \ \ \) \(\displaystyle = p_6\# + 1\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1\) \(\displaystyle = 30031\)

This sequence is A006862 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).