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
\(\ds 2 \#\) | \(=\) | \(\ds 1 \times 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2\) | ||||||||||||
\(\ds 3 \#\) | \(=\) | \(\ds 2 \times 3\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 6\) | ||||||||||||
\(\ds 5 \#\) | \(=\) | \(\ds 2 \times 3 \times 5\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5 \times \paren {2 \times 3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 5 \times 6\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 30\) | ||||||||||||
\(\ds 7 \#\) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 7} \times \paren {3 \times 5}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 14 \times 15\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 210\) | ||||||||||||
\(\ds 17 \#\) | \(=\) | \(\ds 2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \times 3 \times 7 \times 17} \times \paren {5 \times 11 \times 13}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 714 \times 715\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 510 \, 510\) |
$\blacksquare$
Sources
- 1974: C. Nelson, D.E. Penney and C. Pomerance: 714 and 715 (J. Recr. Math. Vol. 7, no. 2: pp. 87 – 89)
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $714$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $714$