Functions that satisfy $f(n) = \sum_{d|n, d\neq n} f(d)$ and $f(1) = 1$ Let $f: N \rightarrow N$ satisfies $$f(n) = \sum_{d|n, d\neq n} f(d)$$ and $f(1) = 1$
Compute $$\sum_{k=0}^{\infty} \frac{f(2021^k)}{2021^k}$$
The function is curious. I found https://en.wikipedia.org/wiki/M%C3%B6bius_function, but it doesn't quite help me..
 A: The first step is to identify the sequence defined by
$$ f(n) = \sum_{d|n,d\ne n}f(d) \quad \text{ and } \quad f(1)=1 \tag{1}. $$
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 $\, N := 2021 = 43\cdot 47 \,$ 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 $\,S\approx 1.00149080995973599729777313. $
