For a natural number $n\in\mathbb N$, consider the sum of the squares of all its digits and denote it by $S^2(n)$. Let $T_0(n)=n$, $T_1(n)=S^2(n)$, \dots, $T_{k+1}(n)=T_1(T_k(n))$ for $k\ge 1$. The number $n$ is called lucky if there exists $k\ge 1$ such thatFor a natural number $n\in\mathbb N$, consider the sum of the squares of all its digits and denote it by $S^2(n)$. Let $T_0(n)=n$, $T_1(n)=S^2(n)$, \dots, $T_{k+1}(n)=T_1(T_k(n))$ for $k\ge 1$. The number $n$ is called lucky if there exists $k\ge 1$ such that $T_k(n)=1$. Otherwise, the number $n$ is called unlucky. It is known that for every unlucky number $n$ there exists $k\ge 1$ such that $T_k(n)\in C=\{4,16,37,58,89,145,42,20\}$. If $c\in C$, then we say that the unlucky number $n$ is $c$-unlucky if $T_k(n)=c$ and $T_{k-1}(n)\not\in C$ for some $k\ge 1$. In this paper, the density of $c$-unlucky numbers is investigated. Estimates for the upper and lower asymptotic densities of $c$-unlucky numbers are obtained and it is proved that there is no natural density of unlucky numbers.

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