ll cal(ll x, ll lim)
{
    ll res = x;
    do {
        int t = x % 10;
        if (res > lim / t) return -1; /// (x * f(x) > N)
        res *= t;
    } while (x /= 10);
    return res;
}

ll brute(ll n)
{
    ll res = 0;
    for (ll x = 1; x <= n; ++x)
        res += cal(x, n) > 0; /// 1 <= x * f(x) <= N

    return res;
}

/// Run 2e13 under 1 secs in (-O2) flag
/// X: current X
/// Y: f(X)
/// T: X * f(X)

ll N;
int res = 0;
vector<int> d;
void build(ll X = 0, ll Y = 1, ll T = 0)
{
    if (T >= 1) ++res; /// 1 <= x * f(x) <= N
    for (int v = 1; v <= 9; ++v) /// if (v = 0) then f(x) = 0
    {
        ll NX = X * 10 + v; /// Insert rightmost digits
        ll NY = Y * v;      /// Calculate digits production
        ll NT = NX * NY;
        if (NT > N) break;  /// x * f(x) > N
        build(NX, NY, NT);
    }
}   

/// [L, R] = [1, N]
/// lim = sqrt(N)
ll solve(int p2 = 0, int p3 = 0, int p5 = 0, int p7 = 0, ll P = 1)
{
    if (P > lim) return 0; /// Dont care such tuples whose P > sqrt(N)

    VL = (L + val - 1) / val; ///  ceil(L / P)
    VR = R / val;             /// floor(R / P)
    ll res = magic(0, 18, p2, p3, p5, p7); /// Calculating subproblem

    /// By doing these if-condition, it is guarantee that all tuples generated are all unique
    if (!p3 && !p5 && !p7) res += solve(p2 + 1, p3, p5, p7, P * 2); /// Continue increasing a
    if (       !p5 && !p7) res += solve(p2, p3 + 1, p5, p7, P * 3); /// Continue increasing b
    if (              !p7) res += solve(p2, p3, p5 + 1, p7, P * 5); /// Continue increasing c
                           res += solve(p2, p3, p5, p7 + 1, P * 7); /// Continue increasing d

    return res;
}

• $$$VL, VR$$$ means the range of $$$x$$$ needed to get $$$1 \leq x \times f(x) \leq N$$$
• Current prefix of building number is $$$X$$$
• We $$$N$$$ digits left to build to the right of $$$X$$$
• $$$p2, p3, p5, p7$$$ means the left factors in $$$f(x)$$$
• $$$min(X, N) = X \times 10^N = \overline{X\underbrace{00000\dots}_{N}}$$$ means the smallest possible number with prefix $$$X$$$ and $$$N$$$ left digits
• $$$max(X, N) = X \times 10^N + 10^N - 1 = \overline{X\underbrace{99999\dots}_{N}}$$$ means the greatest possible number with prefix $$$X$$$ and $$$N$$$ left digits
• $$$cost[v][p]$$$ means greatest divisor $$$p^k$$$ in $$$v$$$ is when $$$k = cost[v][p]$$$
• $$$memo$$$ means current $$$x$$$ is in range $$$[VL, VR]$$$ so we can use memorization
• $$$save = f[N][p2][p3][p5][p7]$$$ means dp-table the numbers of valid number $$$X$$$ with current states
• $$$l_x$$$ means maximum $$$k$$$ that $$$x^k \leq 10^{18}$$$. Where $$$l2, l3, l5, l7, l10$$$ are such those variables we use
• $$$pw10[x] = 10^x$$$

• Initially, $$$X = 0$$$, $$$N = 18$$$, $$$(p2, p3, p5, p7)$$$ is such tuple generated by above function
• $$$X = 0$$$ means it is not build yet
• $$$VL \leq X \leq VR$$$ means $$$1 \leq x \times f(x) \leq N$$$
• $$$N = 0$$$ means $$$X$$$ is built completely
• $$$p2 = p3 = p5 = p7 = 0$$$ means $$$f(x) = P$$$
• $$$max(X) < VL$$$ means $$$x \times f(x) < L$$$ always satisfy which is not desired $$$x$$$ we are considering
• $$$min(X) > VR$$$ means $$$x \times f(x) > R$$$ always satisfy which is not desired $$$x$$$ we are considering
• Building such digit $$$v$$$ will reduce each $$$p2, p3, p5, p7$$$ by $$$cost[v][2], cost[v][3], cost[v][5], cost[v][7]$$$
• $$$memo = false$$$ means if we take memorization here, it is not guarantee that $$$1 \leq x \times f(x) \leq N$$$
• $$$save = -1$$$ means this dp-state it is not calculated yet

const int l2 = 60, l3 = 37, l5 = 25, l7 = 21, l10 = 19;
ll        pw2[l2], pw3[l3], pw5[l5], pw7[l7], pw10[l10];
int       cost[10][10];
void precal()
{
    pw2[0] = pw3[0] = pw5[0] = pw7[0] = pw10[0] = 1; 
    for (int i2  = 1; i2  < l2 ; ++i2 ) pw2 [i2]  =  2 * pw2 [i2  - 1];
    for (int i3  = 1; i3  < l3 ; ++i3 ) pw3 [i3]  =  3 * pw3 [i3  - 1];
    for (int i5  = 1; i5  < l5 ; ++i5 ) pw5 [i5]  =  5 * pw5 [i5  - 1];
    for (int i7  = 1; i7  < l7 ; ++i7 ) pw7 [i7]  =  7 * pw7 [i7  - 1];
    for (int i10 = 1; i10 < l10; ++i10) pw10[i10] = 10 * pw10[i10 - 1];
                                                                                /// 2^a * 3^b * 5^c * 7^d = val
    cost[1][2] = 0;    cost[1][3] = 0;    cost[1][5] = 0;    cost[1][7] = 0;    ///  0     0     0     0  =  1
    cost[2][2] = 1;    cost[2][3] = 0;    cost[2][5] = 0;    cost[2][7] = 0;    ///  1     0     0     0  =  2
    cost[3][2] = 0;    cost[3][3] = 1;    cost[3][5] = 0;    cost[3][7] = 0;    ///  0     1     0     0  =  3
    cost[4][2] = 2;    cost[4][3] = 0;    cost[4][5] = 0;    cost[4][7] = 0;    ///  2     0     0     0  =  4
    cost[5][2] = 0;    cost[5][3] = 0;    cost[5][5] = 1;    cost[5][7] = 0;    ///  0     0     1     0  =  5
    cost[6][2] = 1;    cost[6][3] = 1;    cost[6][5] = 0;    cost[6][7] = 0;    ///  1     1     0     0  =  6
    cost[7][2] = 0;    cost[7][3] = 0;    cost[7][5] = 0;    cost[7][7] = 1;    ///  0     0     0     1  =  7
    cost[8][2] = 3;    cost[8][3] = 0;    cost[8][5] = 0;    cost[8][7] = 0;    ///  3     0     0     0  =  8
    cost[9][2] = 0;    cost[9][3] = 2;    cost[9][5] = 0;    cost[9][7] = 0;    ///  0     2     0     0  =  9
}

ll VL, VR;
ll f[l10][l2][l3][l5][l7];
ll magic(ll X, int N, int p2, int p3, int p5, int p7)
{
    if (p2 < 0 || p3 < 0 || p5 < 0 || p7 < 0) return 0;
    if (N == 0) return (p2 + p3 + p5 + p7 == 0) && (VL <= X && X <= VR);

    ll mn = X * pw10[N];
    ll mx = mn + pw10[N] - 1;
    if (mx < VL || mn > VR) return 0;

    ll &save = f[N][p2][p3][p5][p7];
    bool memo = (VL <= mn) && (mx <= VR);
    if (memo) if (save != -1) return save;
    
    ll res = 0;
    if (X == 0) res = magic(0, N - 1, p2, p3, p5, p7);
    for (int v = 1; v <= 9; ++v)
    {
        int c2 = cost[v][2];
        int c3 = cost[v][3];
        int c5 = cost[v][5];
        int c7 = cost[v][7];
        res += magic(X * 10 + v, N - 1, p2 - c2, p3 - c3, p5 - c5, p7 - c7);
    }
    if (memo) save = res;
    
    return res;
}

With (N = 1) -> Found 1 valid tuples || upper_bound 1
With (N = 10) -> Found 3 valid tuples || upper_bound 3
With (N = 100) -> Found 10 valid tuples || upper_bound 10
With (N = 1000) -> Found 22 valid tuples || upper_bound 24
With (N = 10000) -> Found 46 valid tuples || upper_bound 50
With (N = 100000) -> Found 83 valid tuples || upper_bound 95
With (N = 1000000) -> Found 141 valid tuples || upper_bound 166
With (N = 10000000) -> Found 225 valid tuples || upper_bound 269
With (N = 100000000) -> Found 338 valid tuples || upper_bound 414
With (N = 1000000000) -> Found 493 valid tuples || upper_bound 612
With (N = 10000000000) -> Found 694 valid tuples || upper_bound 874
With (N = 100000000000) -> Found 951 valid tuples || upper_bound 1212
With (N = 1000000000000) -> Found 1273 valid tuples || upper_bound 1640
With (N = 10000000000000) -> Found 1670 valid tuples || upper_bound 2172
With (N = 100000000000000) -> Found 2155 valid tuples || upper_bound 2825
With (N = 1000000000000000) -> Found 2736 valid tuples || upper_bound 3614
With (N = 10000000000000000) -> Found 3427 valid tuples || upper_bound 4558
With (N = 100000000000000000) -> Found 4246 valid tuples || upper_bound 5676
Error: 5701

With (N = 1) -> Found 1 valid tuples || upper_bound 1
With (N = 2) -> Found 1 valid tuples || upper_bound 1
With (N = 4) -> Found 2 valid tuples || upper_bound 2
With (N = 8) -> Found 2 valid tuples || upper_bound 3
With (N = 16) -> Found 4 valid tuples || upper_bound 4
With (N = 32) -> Found 5 valid tuples || upper_bound 6
With (N = 64) -> Found 8 valid tuples || upper_bound 8
With (N = 128) -> Found 10 valid tuples || upper_bound 11
With (N = 256) -> Found 14 valid tuples || upper_bound 14
With (N = 512) -> Found 17 valid tuples || upper_bound 19
With (N = 1024) -> Found 23 valid tuples || upper_bound 24
With (N = 2048) -> Found 28 valid tuples || upper_bound 30
With (N = 4096) -> Found 36 valid tuples || upper_bound 38
With (N = 8192) -> Found 43 valid tuples || upper_bound 47
With (N = 16384) -> Found 53 valid tuples || upper_bound 58
With (N = 32768) -> Found 63 valid tuples || upper_bound 71
With (N = 65536) -> Found 77 valid tuples || upper_bound 85
With (N = 131072) -> Found 89 valid tuples || upper_bound 102
With (N = 262144) -> Found 106 valid tuples || upper_bound 121
With (N = 524288) -> Found 122 valid tuples || upper_bound 143
With (N = 1048576) -> Found 143 valid tuples || upper_bound 167
With (N = 2097152) -> Found 164 valid tuples || upper_bound 195
With (N = 4194304) -> Found 190 valid tuples || upper_bound 225
With (N = 8388608) -> Found 216 valid tuples || upper_bound 260
With (N = 16777216) -> Found 248 valid tuples || upper_bound 298
With (N = 33554432) -> Found 279 valid tuples || upper_bound 339
With (N = 67108864) -> Found 317 valid tuples || upper_bound 386
With (N = 134217728) -> Found 355 valid tuples || upper_bound 437
With (N = 268435456) -> Found 400 valid tuples || upper_bound 492
With (N = 536870912) -> Found 446 valid tuples || upper_bound 553
With (N = 1073741824) -> Found 498 valid tuples || upper_bound 620
With (N = 2147483648) -> Found 553 valid tuples || upper_bound 692
With (N = 4294967296) -> Found 614 valid tuples || upper_bound 770
With (N = 8589934592) -> Found 679 valid tuples || upper_bound 855
With (N = 17179869184) -> Found 749 valid tuples || upper_bound 946
With (N = 34359738368) -> Found 825 valid tuples || upper_bound 1045
With (N = 68719476736) -> Found 905 valid tuples || upper_bound 1152
With (N = 137438953472) -> Found 991 valid tuples || upper_bound 1266
With (N = 274877906944) -> Found 1083 valid tuples || upper_bound 1388
With (N = 549755813888) -> Found 1181 valid tuples || upper_bound 1520
With (N = 1099511627776) -> Found 1286 valid tuples || upper_bound 1660
With (N = 2199023255552) -> Found 1398 valid tuples || upper_bound 1810
With (N = 4398046511104) -> Found 1517 valid tuples || upper_bound 1970
With (N = 8796093022208) -> Found 1646 valid tuples || upper_bound 2140
With (N = 17592186044416) -> Found 1780 valid tuples || upper_bound 2321
With (N = 35184372088832) -> Found 1924 valid tuples || upper_bound 2513
With (N = 70368744177664) -> Found 2074 valid tuples || upper_bound 2717
With (N = 140737488355328) -> Found 2235 valid tuples || upper_bound 2933
With (N = 281474976710656) -> Found 2402 valid tuples || upper_bound 3161
With (N = 562949953421312) -> Found 2581 valid tuples || upper_bound 3403
With (N = 1125899906842624) -> Found 2767 valid tuples || upper_bound 3658
With (N = 2251799813685248) -> Found 2966 valid tuples || upper_bound 3928
With (N = 4503599627370496) -> Found 3174 valid tuples || upper_bound 4212
With (N = 9007199254740992) -> Found 3393 valid tuples || upper_bound 4511
With (N = 18014398509481984) -> Found 3625 valid tuples || upper_bound 4826
With (N = 36028797018963968) -> Found 3866 valid tuples || upper_bound 5157
With (N = 72057594037927936) -> Found 4121 valid tuples || upper_bound 5505
With (N = 144115188075855872) -> Found 4386 valid tuples || upper_bound 5870
With (N = 288230376151711744) -> Found 4665 valid tuples || upper_bound 6253
With (N = 576460752303423488) -> Found 4955 valid tuples || upper_bound 6655
With (N = 1152921504606846976) -> Found 5260 valid tuples || upper_bound 7076
Error: 23112

#include <iostream>
#include <cmath>
#include <cstdio>

using namespace std;

typedef long long ll;

ll L;
ll R;
int lim;
ll solve(int p2 = 0, int p3 = 0, int p5 = 0, int p7 = 0, ll val = 1)
{
    if (val > lim) return 0;

    ll res = 1;
    if (!p3 && !p5 && !p7) res += solve(p2 + 1, p3, p5, p7, val * 2);
    if (       !p5 && !p7) res += solve(p2, p3 + 1, p5, p7, val * 3);
    if (              !p7) res += solve(p2, p3, p5 + 1, p7, val * 5);
                           res += solve(p2, p3, p5, p7 + 1, val * 7);

    return res;
}

const double EPS = 0.0188704;
int main()
{
    freopen("Test.OUT", "w", stdout);

    double error = 0;
    ll mul = 2;
    ll over = (1LL << 62) / mul;
    for (ll t = 1; t < over; t *= mul)
    {
        L = 1;
        R = t;
        lim = sqrt(R);
        int res = solve();
        double base = log(sqrt(sqrt(R))) + 1;
        int complexity = ceil(EPS + (base / log(2)) * (base / log(3)) * (base / log(5)) * (base / log(7)));

        cout << "With (N = " << R << ") -> Found " << res << " valid tuples || upper_bound " << complexity << endl;
        int delta = complexity - res;
        if (delta < 0)
        {
            cout << "Program failed at (t = " << t << ") -> ";
            cout << "Returning " << res << " compared to " << complexity << endl;
            exit(0);
        }
        else error += delta;
    }
    cout << "Error: " << error;

    return 0;
}