# Definition:Euclid Number/Sequence

## 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).