Consecutive Integers whose Product is Primorial
Jump to navigation
Jump to search
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
- 1974: C. Nelson, D.E. Penney and C. Pomerance: 714 and 715 (J. Recr. Math. Vol. 7, no. 2: 87 – 89)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $714$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $714$