Take the sequence $S$ to be $4, 3, 8, 5, 12, 7, 16, 9, 20, 11, 24, 13,...$. Clearly the odd indices of the sequence are elements of $4\mathbb{N}^+$, so the odd indices of $S$ form a group without inverses or the identity element. The even indices of the sequence are the odd numbers greater than 1.

We have $S_a + S_b -1 = S_{a+b}$ where $a, b$ are even indices of $S$. I am trying to find a more general set of operations/identities over $S$ such that $S_x \star S_y = S_{f(x, y)}$ for any indices $x, y$ for example. Is this possible? How would I go about defining something like this?

I am learning about groups/rings/fields right now and trying to apply what I am learning to this, so I am wondering how this problem connects to this area, if it even does.

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    $\begingroup$ @Shaun $a, b$ are even indices. $\endgroup$ – Grassi Apr 21 at 0:16
  • $\begingroup$ This is not clear. Since, say, the even numbered terms are just the odd numbers starting from $3$, so long as you stay within the odd numbers you can define $f(a,b)$ by the relation you specify. Similar comment for the odd numbered terms so long as you stay in multiples of $4$. What are you hoping for? $\endgroup$ – lulu Apr 21 at 0:36
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    $\begingroup$ And, just to say, referring to "a group without inverses or identity" is like referring to a bicycle with no wheels or frame. I think all you mean here is a subset of the natural numbers which is closed under addition. $\endgroup$ – lulu Apr 21 at 0:40
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    $\begingroup$ "a group without inverses or the identity" is called a "semigroup"... $\endgroup$ – Arturo Magidin Apr 21 at 0:44

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