Functions that satisfy $f(n) = sum_{d|n, dneq n} f(d)$ and $f(1) = 1$

The first step is to identify the sequence defined by $$ f(n) = sum_{d|n,dne n}f(d) quad text{ and } quad f(1)=1 tag{1}. $$ The first terms are $,1, 1, 1, 2, 1, 3, 1, 4, 2, 3,dots,$ and an OEIS search finds it to be A074206 "Kalmár's [Kalmar's] problem: number of ordered factorizations of n." Notice that $,f(n),$ only depends on the form of the factorization of $,n,$ into primes. Thus, $,f(p^n)=2^{n-1},$ if $,p,$ is any prime. The next step would be to find $,U(n,m):=f(p^n q^m),$ where $,p,q,$ are two distinct primes. This is given by A059576 "Summatory Pascal triangle T(n,k) (0 <= k <= n) read by rows." More precisely, $$ T(n,k)=U(n-k,k) tag{2} $$ where $,T,$ is the entries of the infinite $2$D array $,U,$ read by anti-diagonals. The OEIS entry gives the two-variable generating function $$ sum_{n=0}^infty sum_{m=0}^infty U(n,m), z^n w^m = frac{(1-z) (1-w)}{(1 - 2 w - 2 z + 2 z w)}. tag{3} $$

Now we want to find the infinite sum $$ S := sum_{k=0}^infty frac{f(2021^k)}{2021^k}. tag{4} $$ Notice that $, N := 2021 = 43cdot 47 ,$ is the product of two primes. Thus, $$ S = sum_{k=0}^infty frac{U(k,k)}{N^k} = F(1/N) quad text{ where } quad F(x) := sum_{k=0}^infty U(k,k),x^k. tag{5} $$ Similar to the case of OEIS sequence A008288 "Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals." and its main diagonal OEIS sequence A001850 "Central Delannoy numbers" we find that $,U(k,k),$ is OEIS sequence A052141 and its generating function $,y=F(x),$ satisfies $$ 0 = (4x^2-12x+1)(y^2-y)+(x^2-3x). tag{6} $$ The solution is $$ y = frac{ 1-12x+4x^2+sqrt{1-12x+4x^2}}{2-24x+8x^2}. tag{7} $$ Substituting $,x = 1/N,$ gives $,Sapprox 1.00149080995973599729777313. $