The Scholz conjecture on addition chain is true for infinitely many integers with $\mathrm{\ell }(2n)=\mathrm{\ell }(n)$

It is known that the Scholz conjecture on addition chains is true for all integers $n$ with $\mathrm{\ell }(2n)=\mathrm{\ell }(n)+1$. There exists infinitely many integers with $\mathrm{\ell }(2n)\le \mathrm{\ell }(n)$ and we don’t know if the conjecture still holds for them. The conjecture is also proven to hold for integers $n$ with $v(n)\le 5$ and for infinitely many integers with $v(n)=6$. There is no specific results on integers with $v(n)=7$. In \cite{thurber}, an infinite list of integers satisfying $\mathrm{\ell }(n)=\mathrm{\ell }(2n)$ and $v(n)=7$ is given. In this paper, we prove that the conjecture holds for all of them.