Pairs of Consecutive Integers with 6 Divisors
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Theorem
The following sequence of integers are those $n$ which fulfil the equation:
- $\map {\sigma_0} n = \map {\sigma_0} {n + 1} = 6$
where $\map {\sigma_0} n$ denotes the divisor count function.
That is, they are the first of pairs of consecutive integers which each have $6$ divisors:
- $44, 75, 98, 116, 147, 171, 242, 243, 244, 332, \ldots$
This sequence is A049103 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Proof
From Divisor Count Function from Prime Decomposition:
- $\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$
where:
- $r$ denotes the number of distinct prime factors in the prime decomposition of $n$
- $k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.
\(\ds \map {\sigma_0} {44}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $44$ | |||||||||||
\(\ds \map {\sigma_0} {45}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $45$ |
\(\ds \map {\sigma_0} {75}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $75$ | |||||||||||
\(\ds \map {\sigma_0} {76}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $76$ |
\(\ds \map {\sigma_0} {98}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $98$ | |||||||||||
\(\ds \map {\sigma_0} {99}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $99$ |
\(\ds \map {\sigma_0} {116}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $116$ | |||||||||||
\(\ds \map {\sigma_0} {117}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $117$ |
\(\ds \map {\sigma_0} {147}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $147$ | |||||||||||
\(\ds \map {\sigma_0} {148}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $148$ |
\(\ds \map {\sigma_0} {171}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $171$ | |||||||||||
\(\ds \map {\sigma_0} {172}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $172$ |
\(\ds \map {\sigma_0} {242}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $242$ | |||||||||||
\(\ds \map {\sigma_0} {243}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $243$ | |||||||||||
\(\ds \map {\sigma_0} {244}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $244$ | |||||||||||
\(\ds \map {\sigma_0} {245}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $245$ |
\(\ds \map {\sigma_0} {332}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $332$ | |||||||||||
\(\ds \map {\sigma_0} {333}\) | \(=\) | \(\ds 6\) | $\sigma_0$ of $333$ |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $44$