These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:

Context:

Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring by observing that each divisor $d$ has

$$0 \le v_p(d) \le v_p(n)$$

Hence we can add two divisors $d,e$ by setting:

$$d \oplus e := \prod_{p | n} p^{v_p(d)+v_p(e) \mod (v_p(n)+1)}$$

and similarily we can multiply them by setting: $$d \otimes e := \prod_{p | n} p^{v_p(d) \cdot v_p(e) \mod (v_p(n)+1)}$$

Then, if $n = p_1^{a_1} \cdots p_r^{a_r}$, this ring will be isomorphic to the ring

$$\mathbb{Z}/(a_1+1) \times \cdots \times \mathbb{Z}/(a_r+1)$$

If $n$ is squarefree, than this reduces to :

$$d\oplus e = \frac{de}{\gcd(d,e)^2}$$

$$d\otimes e = \gcd(d,e)$$

Both methods rely on the character tables of abelian groups of order $2^r$ and on Dedekind group matrices defined over the set of divisors or unitary divisors of $n$:

Method one:

Let $U(n):=$ set of unitary ($\gcd(d,n/d)=1$) divisors of $n$ ordered by their absolute value:

Then $H_n$ is a Hadamard matrix:

$$H_n := ((-1)^{\omega(\gcd(d,e))})_{d,e \in U(n)}$$

Method two:

Let $X(n):= \{ d : d|n, \gcd(d,n/d)=1, \forall p|d: v_p(d)\equiv 1 \mod(2) \}$.

and let $\chi_n(d,e) := \prod_{p|n} \exp(\frac{2 \pi \sqrt{-1}}{v_p(n)+1})^{v_p(d)v_p(e)}$ Then this matrix is a Hadamard matrix as the character table of some abelian group $\mathbb{Z}/(2)^r$:

$$H^{(2)}_n := (\chi_n(d,e))_{d,e \in X(n)}$$

Example computations, show, that these two construction are in general not the same and they are not the same as the induction construction given at Wikipedia:

SageMath-Computations

Are these constructions known or are they maybe equivalent in some sense to the inductive Sylvester construction given at Wikipedia?

Thanks for your help!

These two observations came while researching the empty set of odd perfect numbers and unitary perfect numbers:

Context:

Let $n$ be a natural number and $D_n$ be the set of divisors. We can make this set to a ring by observing that each divisor $d$ has

$$0 \le v_p(d) \le v_p(n)$$

Hence we can add two divisors $d,e$ by setting:

$$d \oplus e := \prod_{p | n} p^{v_p(d)+v_p(e) \mod (v_p(n)+1)}$$

and similarily we can multiply them by setting: $$d \otimes e := \prod_{p | n} p^{v_p(d) \cdot v_p(e) \mod (v_p(n)+1)}$$

Then, if $n = p_1^{a_1} \cdots p_r^{a_r}$, this ring will be isomorphic to the ring

$$\mathbb{Z}/(a_1+1) \times \cdots \times \mathbb{Z}/(a_r+1)$$

If $n$ is squarefree, than this reduces to :

$$d\oplus e = \frac{de}{\gcd(d,e)^2}$$

$$d\otimes e = \gcd(d,e)$$

Both methods rely on the character tables of abelian groups of order $2^r$ and on Dedekind group matrices defined over the set of divisors or unitary divisors of $n$:

Method one:

Let $U(n):=$ set of unitary ($\gcd(d,n/d)=1$) divisors of $n$ ordered by their absolute value:

Then $H_n$ is a Hadamard matrix:

$$H_n := ((-1)^{\omega(\gcd(d,e))})_{d,e \in U(n)}$$

Method two:

Let $X(n):= \{ \sqrt{\operatorname{rad}(d)d} : d|n, \gcd(d,n/d)=1, \forall p|d: v_p(d)\equiv 1 \mod(2) \}$.

and let $\chi_n(d,e) := \prod_{p|n} \exp(\frac{2 \pi \sqrt{-1}}{v_p(n)+1})^{v_p(d)v_p(e)}$ Then this matrix is a Hadamard matrix as the character table of some abelian group $\mathbb{Z}/(2)^r$:

$$H^{(2)}_n := (\chi_n(d,e))_{d,e \in X(n)}$$

Example computations, show, that these two construction are in general not the same and they are not the same as the induction construction given at Wikipedia:

SageMath-Computations

Are these constructions known or are they maybe equivalent in some sense to the inductive Sylvester construction given at Wikipedia?

Thanks for your help!