What is the type of a constructor?

I know constructor do not have a return type. Though I wonder, what is the type of a constructor then? Does a constructor have a type?

I tried this

struct A { A() {} };

template <typename A> struct foo;

int main() { foo< decltype(&A::A) > f; }

to get the error (gcc)

prog.cc: In function 'int main()':
prog.cc:5:32: error: taking address of constructor 'constexpr A::A(A&&)'
    5 | int main() { foo< decltype(&A::A) > f; }
      |                                ^
prog.cc:5:35: error: template argument 1 is invalid
    5 | int main() { foo< decltype(&A::A) > f; }
      |                                   ^

...well, ok I cannot take the address. Also this fails:

int main() { foo< decltype(A::A) > f; }

with

prog.cc: In function 'int main()':
prog.cc:5:32: error: decltype cannot resolve address of overloaded function
    5 | int main() { foo< decltype(A::A) > f; }
      |                                ^
[...]

which is probably just a very confusing error message caused by the same reason as above (cannot take adress of constructor) and I dont know what else to try..

What is the type of a constructor?

If it has no type, then what is it? Certainly it is not A (member_function)().

PS: To clarify what is my confusion: cpprefernce states

Constructor is a special non-static member function of a class that is used to initialize objects of its class type.

And my logic goes like this: Member function have a type, constructors are special kinds of member functions, hence they should have a type. I know the reasoning is flawed, but why?

I think these quotes from the C++ 17 Standard will be relevant (15.1 Constructors)

1 Constructors do not have names....

and

2 A constructor is used to initialize objects of its class type. Because constructors do not have names, they are never found during name lookup; however an explicit type conversion using the functional notation (8.5.1.3) will cause a constructor to be called to initialize an object. [ Note: For initialization of objects of class type see 15.6. — end note ]

and

10 A return statement in the body of a constructor shall not specify a return value. The address of a constructor shall not be taken.