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I have only seen か(個/カ/ヶ) in か月、か年(五か年計画)、か国(六か国会議)、か所、か条.

Is it a productive component which can be used in making other new measure words?

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No, it’s not. You cannot coin a new counter like this.

EDIT: か月, か国 and か所 work as ordinary counters in contemporary Japanese, but others usually appear in historical terms (e.g. 五箇伝) or fixed phrases (e.g., 三が日). If you are working on historical fiction or fantasy works, you might use this pattern to create a new archaic-sounding term for use within the work. For example, something like 三箇龍玉 or 三ヶ龍玉 (Trinity Dragon Orbs) seems reasonable to me as a keyword in a fantasy work, even though Google does not know a single example of this. Still, this is a very rare convention, and I doubt this would be considered "productive" in the linguistic sense.

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If such a mechanism were to exist, then it would be something like:

3カ国語 -> 3カ(の)国語

We understand the hypothesis, but it turns out to be false because we can't create new counters (we can't go the other way around).

三個の青あざ -> 三個青あざ (X)

Because NUMBER個青あざ should be a noun, but of course it isn't.

So the questions shifted from "how to make counters" to "how to make compound nouns". They just work conventionally and are spontaneously created as any many other words.

The NUMBERか[月/年/etc] counter tends to be only used as nouns and not as adverbs. So omitting our hypothetical の would mean that we didn't need that function in the first time.

I mean this:

猫の三匹 (as a noun) & 猫が三匹いる (as an adverb)

We can't say: 五ヶ[月/年/etc]+VERB

They are sort of a different breed.

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