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][1]

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

Thanks for your help!

  [1]: https://sagecell.sagemath.org/?z=eJyNVMlu2zAQvQfIPzAuCpAq05i8BChA9Kr-QgTBUE3GZU2TrCQb-fwOF22W01Q-GDN6fMuMJKle0f6XxpZKRiUn3-7vEFweCeRbJ3GF1JvHvPC6-FFcGnNueu0sBrAn85pD_YSnhoX6CyMEvboW2LQFOn1SO6kvunNthy2pSdLS4MDZ3-dD0yvsiRA-mwhXq_pza9HLC9xJXWU6tQZsU-f-TkIgqUzfhEhDHJBojME5aMj5ILbRmuTB28wVWXOzD4TDL8rqS8eBIoMyoEIyKV0JBVOTUSFYnQIkLt3tYPA7J-WKMCSplpP-zOH8B6NOvG2zJoyLXp0h81jrVLCR7k_b40AoSSHzqm-lPOwBQu1TTIkaK-fpJKkHoVPTs0nIOXgGs_ZCGmCtfsNVNaxTzaSdq2OhckFm5LPdnC2QV-8vZmG5vi2P8CMjRWGUvR5eOq3IfCZnOzMGxWisLCdboGyF2K6fslGU1fU_34Ofxu2PuxEeyMElzf81HTuPQys5CdZssNY29qAwo-x5_BaEj4HRXZ9iwojCi53u2fE74X1l6qLABm7GoGZiiyPyZHzjSwbH8rZT5xQ6uGRFyb72cKjzrlOYkGpLt3n8IG57vLEUdhXhG4osPbHMcAyt6a04McrJ4mAULfGRke8bWjIhylgMlniyxCdLPFriRcn_yxIfLQ3CR760xFeWeDTBoyMucpExnxIoOr0VZbPoleyaeg25Zon5rzDzmcxw_AYOnP4Fg6ayLw==&lang=sage&interacts=eJyLjgUAARUAuQ==