As you probably know, congruent numbers $N$ and elliptic curves of the form $$E_N:y^2=x^3-N^2x$$ are intimately connected.
While playing around with curves of this form, I found that $E_N$ will have exactly $3$ integral points, precisely the $2$-torsion: $$(\pm N,0),(0,0)$$ when $N$ is NOT congruent (proof of this?). However, there are some numbers $N$ that are congruent yet $E_N$ ALSO has only $3$ integral points. These are: $$13,23,31,37,38,47,52,53,55,61,62,71,79,86,87,92,93,94,95,101,103,109\ldots(?)$$ I couldn't find any reference to this sequence, in OEIS or otherwise. Is there something significant going on here?
I am aware of the Nagell-Lutz theorem, which states that all rational points on an elliptic curve of finite order are integral. But I also know that the converse is not true.
Edit: The SAGE command I was using to calculate integral points burnt out at $N=113$. Out of $57$ congruent numbers, $22$ had exactly $3$ integral points.