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Let $\varphi$ denote Euler's totient function.

QUESTION. Is it true that for each positive integer $k$ large integers $n$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive integers such that $\varphi(n_1),\ldots,\varphi(n_k)$ are $k$th powers?

In 2014, I conjectured that the answer is positive (cf. http://oeis.org/A237123). For example, $$101=1+15+85\ \ \ \text{with}\ \varphi(1)=1^3,\ \varphi(15)=2^3,\ \varphi(85)=4^3.$$

For any positive integer $k$, let $s(k)$ be the least positive integer for which any integer $n\ge s(k)$ can be written as $n_1+\ldots+n_k$ with $n_1,\ldots,n_k$ distinct positive integers such that $\varphi(n_1),\ldots,\varphi(n_k)$ are $k$th powers. My computation suggests that $$s(2)=70640,\ \ s(3)=935,\ \ s(4)=3273.$$

Any ideas towards the solution of my above question? Your comments are welcome!

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  • $\begingroup$ This question essentially asks whether the set of positive integers $n$ with $\varphi(n)$ a $k$th power is an asymptotic additive base of order $k$. $\endgroup$
    – Zhi-Wei Sun
    Commented May 2, 2020 at 3:23
  • $\begingroup$ Are conditional answers accepted? $\endgroup$
    – user156885
    Commented May 5, 2020 at 6:54
  • $\begingroup$ @zz7948 Any progress is welcome! $\endgroup$
    – Zhi-Wei Sun
    Commented May 5, 2020 at 13:32

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