# It's more common than you think

For positive integer $$n$$, define $$M_n$$ as an $$n$$-digit positive integer such that in decimal representation, the last $$n$$ (rightmost) digits of $$M_n^2$$ is $$M_n$$ itself. Denote $$L_n$$ as the sum of all possible values of $$M_n$$. Find the 14th smallest integer $$k$$ that doesn't satisfy the equation $$L_k=10^k+1$$.

Details and Assumptions:

• As an explicit example: If $$n=3$$, then $$M_3 = 376,625$$ because $$376^2=141\underline{376}$$ and $$625^2=390\underline{625}$$ and then $$L_3=10^3+1$$ but if $$n=4$$, $$M_4=9376$$ only, so $$L_4\ne10^4+1$$.

• I don't consider $$k=1$$ as the smallest integer $$k$$ that doesn't satisfy the equation $$L_k = 10^k+1$$.

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