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In his book "Crystal Sets to Sideband", Frank Harris K0IYE describes using premix oscillators (PMOs) to convert signal from low-frequency LO to a particular ham band.

Many ham bands are harmonically related, so it could be possible to substitute separate crystal oscillators for each ham band with one high frequency crystal oscillator (say, for 10m band) and some kind of frequency divider(s) to generate frequencies for 20m, 40m, 80m and 160m.

Yet I've never seen anything like that. Are there any problems with this scheme?

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Not really a problem – just a technological challenge.

The challenge, in almost any case, lies in the fact that per definition, the input is a harmonic of the output of the frequency divider, and since you don't want to have both the "high" input frequency and the "low" output frequency in your LO signal, you'll have to strongly suppress the frequency of the input signal. Now, the input signal has usually a high chance of being relatively powerful – and hence, hard to suppress.

Now, if you asked me how I'd go about building a flexible LO-generater from a high input frequency, my answer would be:

  1. use one of the many existing LO synthesizer ICs – they work excellently, there's a lot of code that shows how to use them (e.g. attached to some computer via some I²C bridge) and they really get the job done without requiring you to build a lot of specific filters at exotic frequencies, measure them etc. and are far more flexible than just giving you dividers of your input LO; ask e.g. analog.com or Maxim Integrated about fractional-n or integer-n synthesizers: you can often do stuff like taking a single, arbitrary input oscillator, and generate a nearly arbitrary set of output frequencies with one single device. You can probably even get a free sample IC. High-End devices use these ICs, so it's not a "short cut with drawbacks". these ICs do not only allow you to generate non-integer factors of your input frequency, but also non-integer multiples of it – so you could, for example, buy a good 10MHz oscillator, and use it to generate 2.443265 GHz with the accuracy of that 10MHz reference oscillator.
  2. if I didn't feel like just getting a part that was built by someone who knows his trade better than I do, I'd just get some 10€-20€ ARM dev board, or whatever microcontroller platform you'd feel most comfortable with, and used the timer units these devices have to generate a set of already pretty adaptive clocks – for example, I have a ST Nucleo-F446RE board lying around, which I could use to generate a lot of clocks by letting the PWM units count up to values that I can specify via software, at a rate that I specify via software. The oscillator used to make the PWM unit "tick" can be supplied externally, e.g. from a high-quality OCXO if I felt that the simple quartz oscillator on-board of that eval board wasn't sufficiently accurate. These outputs could then either be filtered in an analog fashion (use the excellent Analog Devices Filter Designer ) or used to feed a PLL that controls a voltage-controlled oscillator (VCO) – I'd recommend the later.
  3. Same idea as 2., but build your own control loop: Use a microcontroller with a timer/PWM unit (the ST ARMs tend to have lots of those) to generate a "system tick". Either write a piece of software that counts the rising edges of your your VCO, or use a dedicated integrated piece of hardware (again, the ST ARMs have such counters integrated in their timer units). Compare the number of edges that you counted with the "should-be" value (e.g. in 0.1 milliseconds, your VCO should have produced 750 such edges for a 40m=7.5MHz LO) and increase/decrease the control voltage accordingly.
  4. In any case, I wouldn't use the still-often-thaught analog "regenerative frequency divider" method to build a frequency divider – you'd need pretty good equipment to verify the quality of its operation, and it's not a fun thing to implement right. Also, although they have the advantage of not generating a square wave with the mentioned sinc-shaped spectrum, they tend to have harmonics all over the place unless you can do fantastic filtering (which usually is more expensive/complicated than just getting another quartz).

Yet I've never seen anything like that.

Look harder: Frequency dividers are everywhere in digital hardware, and even more prominently used in radio electronics. They work well, are part of the ICs I mentioned in 1. above, and are extremely easy to implement if you have digital signals – in fact, it's more likely you'll find a frequency divider that does

sine signal -> threshold/Schmitt trigger -> counter -> filter / (PLL+VCO) -> sine signal

than something that works directly on the analog signal[citation needed].

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    $\begingroup$ Clarification: counters used as digital dividers for feedback in a PLL (a system) usually drive the phase frequency detector that produces the frequency adjustment signal that is filtered and used to drive the VCO. They're definitely just about everywhere in digital clock synthesis, though. $\endgroup$ – user2943160 Jun 6 '16 at 2:41

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