C++17 (gcc), 108 bytes

#import<random>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::gcd(n,d);d/=a;}

Try it online!

Same as below, but use C++17's std::gcd.

C++ (gcc), 109 bytes

#import<regex>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::__gcd(n,d);d/=a;}

Try it online!

Because C++ doesn't have native bigint support, this will definitely overflow for n>20.

Require:

• gcc's deprecated import statement.
• gcc's std::__gcd.
• -O0 (I think so) otherwise the compiler will optimize out d/=a.
• At least 64-bit long.

Explanation:

• Let \$d=n!, a=H_n\$.
• Round a*d to nearest integer by casting a*d+.5 to long, and assign to n. Now n/d is the output.
• Simplify the fraction with std::__gcd.

C++17 (gcc) , 108 bytes

Only use integer arithmetic:

#import<random>
int f(int x,long&n,long&d){n=0;d=1;int
a;while(n=n*x+d,d*=x,a=std::gcd(n,d),n/=a,d/=a,--x);}

Try it online!

C++17 (gcc), 108 bytes

#import<random>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::gcd(n,d);d/=a;}

Try it online!

Same as below, but use C++17's std::gcd.

C++ (gcc), 109 bytes

#import<regex>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::__gcd(n,d);d/=a;}

Try it online!

Because C++ doesn't have native bigint support, this will definitely overflow for n>20.

Require:

• gcc's deprecated import statement.
• gcc's std::__gcd.
• -O0 (I think so) otherwise the compiler will optimize out d/=a.
• At least 64-bit long.

Explanation:

• Let \$d=n!, a=H_n\$.
• Round a*d to nearest integer by casting a*d+.5 to long, and assign to n. Now n/d is the output.
• Simplify the fraction with std::__gcd.

C++ (gcc), 109 bytes

#import<regex>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::__gcd(n,d);d/=a;}

Try it online!

Because C++ doesn't have native bigint support, this will definitely overflow for n>20.

Require:

• gcc's deprecated import statement.
• gcc's std::__gcd.
• -O0 (I think so) otherwise the compiler will optimize out d/=a.
• At least 64-bit long.

Explanation:

• Let \$d=n!, a=H_n\$.
• Round a*d to nearest integer by casting a*d+.5 to long, and assign to n. Now n/d is the output.
• Simplify the fraction with std::__gcd.

C++17 (gcc), 108 bytes

#import<random>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::gcd(n,d);d/=a;}

Try it online!

Same as below, but use C++17's std::gcd .

C++ (gcc), 109 bytes

#import<regex>
int f(long&n){double a=0;long
d=1;while(d*=n,a+=1./n,--n);n=a*d+.5;n/=a=std::__gcd(n,d);d/=a;}

Try it online!

Because C++ doesn't have native bigint support, this will definitely overflow for n>20.

Require:

• gcc's deprecated import statement.
• gcc's std::__gcd.
• -O0 (I think so) otherwise the compiler will optimize out d/=a.
• At least 64-bit long.

Explanation:

• Let \$d=n!, a=H_n\$.
• Round a*d to nearest integer by casting a*d+.5 to long, and assign to n. Now n/d is the output.
• Simplify the fraction with std::__gcd.