Smallest Consecutive Even Numbers such that Added to Divisor Count are Equal
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Theorem
$30$ is the smallest positive even integer $n$ with the property:
\(\ds n + \map {\sigma_0} n\) | \(=\) | \(\ds m\) | ||||||||||||
\(\ds \paren {n + 2} + \map {\sigma_0} {n + 2}\) | \(=\) | \(\ds m\) | ||||||||||||
\(\ds \paren {n + 4} + \map {\sigma_0} {n + 4}\) | \(=\) | \(\ds m\) |
where:
- $m \in \Z_{>0}$ is some positive integer
- $\map {\sigma_0} n$ is the divisor count function: the number of divisors of $n$.
In this case, where $n = 30$, we have that $m = 38$.
Proof
From Divisor Count Function from Prime Decomposition, we have:
- $\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where the prime decomposition of $n$ is:
- $n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$
\(\ds 2 + \map {\sigma_0} 2\) | \(=\) | \(\ds 2 + 2\) | \(\ds = 4\) | as $2 = 2^1$ | ||||||||||
\(\ds 4 + \map {\sigma_0} 4\) | \(=\) | \(\ds 4 + 7\) | \(\ds = 7\) | as $4 = 2^2$ | ||||||||||
\(\ds 6 + \map {\sigma_0} 6\) | \(=\) | \(\ds 6 + 4\) | \(\ds = 10\) | as $6 = 2^1 3^1$ | ||||||||||
\(\ds 8 + \map {\sigma_0} 8\) | \(=\) | \(\ds 8 + 4\) | \(\ds = 12\) | as $8 = 2^3$ | ||||||||||
\(\ds 10 + \map {\sigma_0} {10}\) | \(=\) | \(\ds 10 + 4\) | \(\ds = 14\) | as $10 = 2^1 5^1$ | ||||||||||
\(\ds 12 + \map {\sigma_0} {12}\) | \(=\) | \(\ds 12 + 6\) | \(\ds = 18\) | as $12 = 2^2 3^1$ | ||||||||||
\(\ds 14 + \map {\sigma_0} {14}\) | \(=\) | \(\ds 14 + 4\) | \(\ds = 18\) | as $14 = 2^1 7^1$ | ||||||||||
\(\ds 16 + \map {\sigma_0} {16}\) | \(=\) | \(\ds 16 + 5\) | \(\ds = 21\) | as $16 = 2^4$ | ||||||||||
\(\ds 18 + \map {\sigma_0} {18}\) | \(=\) | \(\ds 18 + 6\) | \(\ds = 24\) | as $18 = 2^1 3^2$ | ||||||||||
\(\ds 20 + \map {\sigma_0} {20}\) | \(=\) | \(\ds 20 + 6\) | \(\ds = 26\) | as $20 = 2^2 5^1$ | ||||||||||
\(\ds 22 + \map {\sigma_0} {22}\) | \(=\) | \(\ds 22 + 4\) | \(\ds = 26\) | as $22 = 2^1 11^1$ | ||||||||||
\(\ds 24 + \map {\sigma_0} {24}\) | \(=\) | \(\ds 24 + 8\) | \(\ds = 32\) | as $24 = 2^3 3^1$ | ||||||||||
\(\ds 26 + \map {\sigma_0} {26}\) | \(=\) | \(\ds 26 + 4\) | \(\ds = 30\) | as $26 = 2^1 13^1$ | ||||||||||
\(\ds 28 + \map {\sigma_0} {28}\) | \(=\) | \(\ds 28 + 6\) | \(\ds = 34\) | as $28 = 2^2 7^1$ | ||||||||||
\(\ds 30 + \map {\sigma_0} {30}\) | \(=\) | \(\ds 30 + 8\) | \(\ds = 38\) | as $30 = 2^1 3^1 5^1$ | ||||||||||
\(\ds 32 + \map {\sigma_0} {32}\) | \(=\) | \(\ds 32 + 6\) | \(\ds = 38\) | as $32 = 2^5$ | ||||||||||
\(\ds 34 + \map {\sigma_0} {34}\) | \(=\) | \(\ds 34 + 4\) | \(\ds = 38\) | as $34 = 2^1 17^1$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $38$