Long Solutions of Sequence A348480 of the On-Line Encyclopedia of Integer Sequences

For numbers $x$ coprime to $10$ there exist infinitely many binary numbers $b$ such that the greatest common divisor of $b$ and rev($b$) = $x$ and the sum of digits of $b = x$ (rev($b$) is the digit reversal of $b$). In most cases, the smallest $b$ that fulfill these two constraints contain just a few zeros. But in some cases like for $x = 7, 11, 13$ and $37$, $b$ must contain more zeros than ones and these $b$ are called long solutions. For $11$ and $37$ it follows directly from the fact that these are porous numbers. For $7$ and $13$, the proofs that they have long solutions are presented in this paper.