Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.

Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section A19 "Values of $n$ making $n-2^k$ prime. Odd numbers not of the form $\pm p^a\pm2^b$.", printed p. 67: the question whether some odd integer is not of the form $2^k+s$ with $s$ squarefree, with no example found below $1.4\cdot10^9$. Library home: Guy 2004.

C. Hercher, On the Sum of Squarefree Integers and a Power of Two. arXiv:2411.01964 (2024).