Factorial, Powers and Abelian Groups

Let $f(n)$ denote the number of abelian groups of order $n$ up to isomorphism. Let $m$ and $N$ be fixed positive integers. For several pairs of positive integers $L$ and $R$ such that L \leq R \leq N, find the value of the sum

Let us recall the basic concepts and facts about abelian groups.

An abelian group is a pair G = (A, *), where $A$ is a set and $\ast$ is a binary operation on $A$, that is, for any two elements $a$ and $b$ in $A$, the result a*b is also in $A$. The following properties, called the axioms of an abelian group, must hold:

(i) Associativity: for any $a$, $b$, $c$ in $A$, (a*b)*c = a*(b*c).

(ii) There exists $e$ in $A$ such that for any $a$ in $A$, a*e = e*a = a.

(iii) For any $a$ in $A$, there exists $b$ in $A$ such that a*b = b*a = e.

(iv) Commutativity: for any $a$, $b$ in $A$, a*b = b*a.

An important example of an abelian group is the cyclic group of order $n$: the set of numbers from $0$ to n-1 with the operation of addition modulo $n$. This is denoted as _n.

The direct sum of two abelian groups G = (A, *) and H = (B, \cdot) is the pair G \ H = (C, \times), where C = \{ (a, b) : a \in A,\ b \in B \} and (a_1, b_1) \times (a_2, b_2) = (a_1 * a_2, b_1 \cdot b_2) for all a_1, a_2 \in A and b_1, b_2 \in B.

Two groups G = (A, *) and H = (B, \cdot) are called isomorphic if there is a one-to-one correspondence $f$ from $A$ to $B$ such that f(a_1) \cdot f(a_2) = f(a_1 * a_2) for all a_1, a_2 \in A.

The fundamental theorem of abelian group theory states that any finite abelian group is isomorphic to the direct sum of some cyclic groups.

The Chinese Remainder Theorem states that _{mn} is isomorphic to _m \ _n if and only if $m$ and $n$ are coprime.

The last two facts make it possible to describe all abelian groups of order $n$ up to isomorphism.

For example, if $n$ is prime, then all groups of order $n$ are isomorphic to _n.

There are $2$ non-isomorphic groups of order $4$: _4 and _2 \ _2.

There are $3$ non-isomorphic groups of order $27$: _{27}, _9 \ _3 and _3 \ _3 \ _3.

There are $4$ non-isomorphic groups of order $36$: _4 \ _9, _2 \ _2 \ _9, _4 \ _3 \ _3 and _2 \ _2 \ _3 \ _3.