C++ :

#include <cstdio>
#include <cmath>
/* 位数(x^x) = floor(x * log10(x)) + 1 ；找最小 x 使 >= n */
int main(){
    long long n; if(scanf("%lld", &n) != 1) return 0;
    long long lo = 1, hi = n + 2, ans = -1;
    while(lo <= hi){
        long long mid = lo + (hi - lo) / 2;
        double lg = (double)mid * log10((double)mid);
        long long digits = (long long)floor(lg) + 1;
        if(digits >= n){ ans = mid; hi = mid - 1; }
        else lo = mid + 1;
    }
    printf("%lld\n", ans);
    return 0;
}