I got to thinking about the conditions for a number to be divisible by its totient. It didn't take too long to see the pattern. Figure 1 shows the results for numbers up to 1024.
Figure 1: permalink |
Clearly condition is that the numbers must by of the form \(2^p3^q\) where \(p>0\) and \(q \geq 0\). There are only 35 numbers in this range and, if we extend the range to 10 million, there are still only 178 numbers that satisfy.
These numbers form OEIS A007694:
A007694 | Numbers \(k\) such that \( \phi(k) \) divides \(k\). |
The initial members of the sequence are:
1, 2, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 54, 64, 72, 96, 108, 128, 144, 162, 192, 216, 256, 288, 324, 384, 432, 486, 512, 576, 648, 768, 864, 972, 1024, 1152, 1296, 1458, 1536, 1728, 1944, 2048, 2304, 2592, 2916, 3072, 3456, 3888, 4096, 4374, 4608, 5184, 5832, 6144, 6912, 7776, 8192, 8748, 9216
The numbers must be even, that is they must contain a power of 2. If the numbers are only powers of 3 then the dividend is 1.5. Figure 2 shows the results in the range up to one million. All numbers are of the form \(3^p\) where \(p>0\).
Figure 2: permalink |
What sort of numbers will produce a dividend of 2.5? Well, as it turns out, numbers of the form \(2^p5^q\) where \(p>1\) and \(q>1\). See Figure 3 for the numbers in the range up to one thousand.
Figure 3: permalink |
Figure 4: permalink |
Figure 5: permalink |
Numbers of the from \(2^p11^q23^q\) with \(p>0\), \(q>0\) and \(r>0\) produce a dividend of 2.3. See Figure 6 where the range is up to 100,000.
Figure 6: permalink |
Figure 7: permalink |
Figure 8: permalink |
- \( \dfrac{n}{\phi(n)}=k\) where \(k>0\) if numbers of form \(2^p3^q\) with \(p>0\) and \(q \geq 0\)
- \( \dfrac{n}{\phi(n)}=1.5\) if numbers of form \(3^p\) with \(p>0\)
- \( \dfrac{n}{\phi(n)}=2.2\) if numbers of form \(2^p11^q\) where \(p>0\) and \(q>0\)
- \( \dfrac{n}{\phi(n)}=2.3\) if numbers of form \(2^p11^q23^q\) with \(p>0\), \(q>0\) and \(r>0\)
- \( \dfrac{n}{\phi(n)}=2.5\) if numbers of form \(2^p5^q\) with \(p>0\) and \(q>0\)
- \( \dfrac{n}{\phi(n)}=3.1\) if numbers of form \(2^p3^q31^r\) where \(p>0\), \(q>0\) and \(r>0\)
- \( \dfrac{n}{\phi(n)}=3.3\) if numbers of form \(2^p3^q11^r\) where \(p>0\), \(q>0\) and \(r>0\)
- \( \dfrac{n}{\phi(n)}=3.5\) if numbers of form \(2^p3^q7^r \) with \(p>0\), \(q>0\) and \(r>0\)
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