Consecutive Integers whose Product is Primorial

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Theorem

The following primorials can be expressed as the product of consecutive integers:

$2, 6, 30, 210, 510 \, 510$

This sequence is A161620 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

No others are known.


The corresponding indices of those primorials are:

$2, 3, 5, 7, 17$

This sequence is A215658 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


The corresponding values of $n$ such that $p\# = \paren {n - 1} n$ are:

$2, 3, 6, 15, 715$

This sequence is A215659 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle 2 \#\) \(=\) \(\displaystyle 1 \times 2\)
\(\displaystyle \) \(=\) \(\displaystyle 2\)
\(\displaystyle 3 \#\) \(=\) \(\displaystyle 2 \times 3\)
\(\displaystyle \) \(=\) \(\displaystyle 6\)
\(\displaystyle 5 \#\) \(=\) \(\displaystyle 2 \times 3 \times 5\)
\(\displaystyle \) \(=\) \(\displaystyle 5 \times \paren {2 \times 3}\)
\(\displaystyle \) \(=\) \(\displaystyle 5 \times 6\)
\(\displaystyle \) \(=\) \(\displaystyle 30\)
\(\displaystyle 7 \#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 \times 7} \times \paren {3 \times 5}\)
\(\displaystyle \) \(=\) \(\displaystyle 14 \times 15\)
\(\displaystyle \) \(=\) \(\displaystyle 210\)
\(\displaystyle 17 \#\) \(=\) \(\displaystyle 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {2 \times 3 \times 7 \times 17} \times \paren {5 \times 11 \times 13}\)
\(\displaystyle \) \(=\) \(\displaystyle 714 \times 715\)
\(\displaystyle \) \(=\) \(\displaystyle 510 \, 510\)

$\blacksquare$


Sources