714

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Number

$714$ (seven hundred and fourteen) is:

$2 \times 3 \times 7 \times 17$

The $1$st of the $5$th (and largest known) pair of consecutive integers whose product is a primorial:

$714 \times 715 = 510 \, 510 = 17 \#$

The $8$th integer after $1$, $14$, $30$, $105$, $248$, $264$, $418$ whose divisor sum divided by its Euler $\phi$ value is a square:

$\dfrac {\map {\sigma_1} {714} } {\map \phi {714} } = \dfrac {1728} {192} = 9 = 3^2$

The $10$th positive integer after $1$, $7$, $102$, $110$, $142$, $159$, $187$, $381$, $690$ the sum of whose divisors is a cube:

$\map {\sigma_1} {714} = 1728 = 12^3$

The $21$st integer $n$ after $1$, $3$, $15$, $30$, $35$, $56$, $\ldots$, $210$, $248$, $264$, $357$, $420$, $570$, $616$, $630$ with the property that $\map {\sigma_0} n \divides \map \phi n \divides \map {\sigma_1} n$:

$\map {\sigma_0} {714} = 16$, $\map \phi {714} = 192$, $\map {\sigma_1} {714} = 1728$

Arithmetic Functions on $714$

\(\ds \map {\sigma_0} { 714 }\)

\(=\)

\(\ds 16\)

$\sigma_0$ of $714$

\(\ds \map \phi { 714 }\)

\(=\)

\(\ds 192\)

$\phi$ of $714$

\(\ds \map {\sigma_1} { 714 }\)

\(=\)

\(\ds 1728\)

$\sigma_1$ of $714$

Also see

• Integers whose Ratio between Divisor Sum and Phi is Square

• Previous  ... Next: Consecutive Integers whose Product is Primorial
• Previous  ... Next: Integers whose Ratio between Divisor Sum and Phi is Square
• Previous  ... Next: Numbers such that Divisor Count divides Phi divides Divisor Sum
• Previous  ... Next: Integers whose Divisor Sum is Cube

Historical Note

On April 8, 1974, in Atlanta, Georgia, Henry Aaron hit his $715$th major league homerun, thus eclipsing the previous mark of $714$ long held by Babe Ruth. This event received so much advance publicity that the numbers $714$ and $715$ were on millions of lips. Questions like, 'When do you think he'll get $715$?' were perfectly understood, even with no mention made of Aaron, Ruth or homerun. In all the hubbub it appears certain interesting properties of $714$ and $715$ were overlooked ...

-- C. Nelson, D.E. Penney and C. Pomerance

The whimsical paper so authored was noticed by Paul Erdős. This led to a long and fruitful collaboration between Erdős and Pomerance.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $714$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $714$