I proposed a formula
$a(n,k) = |A-B|=|j^2-2*k^2|, j=(2*n-1),\quad n,k \in N, GCD(j,k)=1,$
for the OEIS series Numbers whose prime factors are all congruent to +1 or -1 modulo 8. $( 1, 7, 17, 23, 31, 41, 47, 49, 71, \cdots )$ that is in limbo because of a disagreement about the series being a function of $1$ variable or $2.\quad$ There is a problem with my formula in that no single variable will generate the entire series if the other variable is fixed.
I guess my questions are:
- Should my formula be dropped?
- Is there as single variable formula for the series?
- Is there an argument in favor of keeping my formula?
I'm open to any ideas.
$\textbf{Update:}$ About the comments and one answer: The formula does not generate all natural numbers of the form $x^2-2y^2.\,$ It generates a subset of natural numbers of the form $(2x-1)^2-2y^2.$
I gather that the formula should probably be dropped because it there is no order in it, any more than there is for complex numbers. The lack or order makes it irrelevant that there are infinite combinations of $\,x,y\,$ that generate each number in the sequence.
It is true that, as Franklin T. Adams-Watters remarks in the first Comment to this sequence, these are the natural numbers of the form $x^2 - 2 y^2$, where $x$ is odd and $x$ and $y$ are relatively prime. That is not the same as saying that the sequence is a function of two variables $x$ and $y$. In fact, the representation as $x^2 - 2 y^2$ is not unique, e.g. $$ 7 = 3^2 - 2 \cdot 1^2 = 5^2 - 2 \cdot 3^2$$