OEIS A212558: Proof of Unproven Conjecture? Proven!

[stored]

Gerald, thanks you for the interesting quiz!

Small addendum about sum of the decimal digits of n.

Let s(n) is the sum of the decimal digits of n.
Consider condition
4*s(n)^2 < n-s(n), (*)
4*s(n)^2 + s(n) < n

Function in left is monotonically increasing function, then
in respect that
s(n) <= 9*log10(n+1)
we get estimation
4*s(n)^2 + s(n) <= 4 * (9 * log10(n+1))^2 + 9*log10(n+1) = 324 * log10(n+1)^2 + 9*log10(n+1)

Maximal solution of the equation
324 * log10(n+1)^2 + 9*log10(n+1) = n
is n0 ~ 4313.68. (I use Wolfram Alpha for getting this value.)
Hence, condition (*) is true for all n > n0.
For n from 2999 to 4313 source statement may be checked by direct computations.