Ancestor's pedigree as a subtree of a descendant's pedigree

Question

Given a person's pedigree as an infinite balanced binary tree of Ahnentafel numbers. These numbers start with 1 as the original person, 2 as their father, 3 as their mother, 4 as their paternal grandfather, etc. I.e. the non-zero positive integers. What is the formula for the subset of Ahnentafel numbers for the subtree (pedigree) of an arbitrary Ahnentafel number (ancestor)? E.g. The result for person 1234 is an increasing set of integers starting with 1234.

Answer 1

Observe that the parents of person $n$ have Ahnentafel numbers $2n$ and $2n+1$. Iterating this observation yields that $A_{n,k}$, the set of ancestors of person $n$ at generation distance $k$, equals $$ \{2^kn,2^kn+1,\ldots,2^k(n+1)-1\}. $$ Thus, the set of ancestors of person $n$ equals $$ A_n=\mathbb Z\cap \bigcup_{k\geq 0}\bigl[2^kn,2^k(n+1)\bigr). $$

Also observe that $m\in A_{n,k}$ if and only if $$ \left\lfloor \frac{m}{2^k}\right\rfloor =n, $$ thus $A_n$ also equals the set of $m\in\mathbb Z$ such that $n$ appears in the sequence $$ \{\lfloor m2^{-k}\rfloor\colon k\geq 0\}. $$