imc

The number $10^n+1$ is divisible by 101 if and only if $10^n\equiv -1(\mathrm{mod}101)$. We note that $(10,10^2,10^3,10^4)\equiv (10,-1,-10,1)(\mathrm{mod}101)$, so the powers of 10 are 4-periodic mod 101. It follows that $10^n\equiv -1(\mathrm{mod}101)$ if and only if $n\equiv 2(\mathrm{mod}4)$. In the given list, $10^2+1,10^3+1,10^4+1,\dots ,10^{2019}+1$, the desired exponents are $2,6,10,\dots ,2018$, and there are $\frac{2020}{4}=\boxed{505}$ numbers in that list.