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

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