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I'm stuck on the following problem:

Find the smallest number $n$ for which the following statement is true: "The number of isomorphism types of abelian groups with $n$ elements is equal to 10". How many such $n$ are there?

My thoughts: I think that the smallest number is $n=144$. How would I prove this? Also, am I correct in saying that there are infinitely many such numbers $n$?

Any help is appreciated!

Sir_Math_Cat
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  • One approach involves a lot of case work around the number of prime factors of $n$. I haven't solved the problem, but it wouldn't surprise me if this happened for all n of the form $n=p^4q^2$ or something similar. – Stella Biderman Mar 27 '17 at 18:58
  • This has been solved in general in the answers, not only for $n=10^6$, or $n=10$. For $n=10$ we have $p(4)p(2)=5\cdot 2=10$, for $144=2^4\cdot 3^2$. – Dietrich Burde Mar 27 '17 at 19:02

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