Let's break down and solve the questions one by one.

Question 3: Finding $g(n)$

Given:

• $n \in \mathbb{N}_0$
• $w(n)$ denotes the number of 1s in the binary representation of $n$.

We need to find a closed formula for $g(n)$ in terms of $n$ and $w(n)$.

Let's analyze the problem by first understanding $g(n)$. In many problems involving binary representations, a common pattern is counting specific bit configurations or analyzing parity.

For example, consider the function $g(n) = n \times 2^{w(n)}$. This function would combine $n$ with the number of 1s in its binary form. Let's now try to apply this and see what happens when we use a specific $n$.

Given $n = 10000000000000011_2$ (in binary):

1. Convert $n$ to decimal: $n = 2^{16} + 2^1 + 2^0 = 65536 + 2 + 1 = 65539$

2. Calculate $w(n)$, the number of 1s in the binary representation: $w(65539) = 3$

3. Let's assume $g(n) = n \times 2^{w(n)}$ as a possible formula. Substituting the values: $g(65539) = 65539 \times 2^3 = 65539 \times 8 = 524312$

4. Convert $524312$ back to binary: $g(65539) = 1000000000010101000_2$

This would be the result, but it does not necessarily match any specific answer format expected. Thus, let's revisit the definition of $g(n)$.

Understanding $g(n)$

Since we are asked to relate $g(n)$ with $w(n)$, consider another simpler function like $g(n) = n \times w(n)$. This will likely involve a modular reduction or additional operation. Let’s try to simplify:

Question 4: Sequence from Binomial Coefficients Modulo 2

Given $n = 1152921504606846987$:

We need to calculate $\binom{n}{k} \mod 2$ for $k = 0, 1, 2, \ldots, 7$.

Steps:

1. Convert $n$ to binary: $n = 1152921504606846987 = 1000000000000000000000000000000011_2$

2. For each $k$, determine $\binom{n}{k} \mod 2$:

   Use Lucas' theorem, which states: $\binom{n}{k} \mod 2 = \prod \binom{n_i}{k_i} \mod 2$ where $n_i$ and $k_i$ are the binary digits of $n$ and $k$, respectively.

3. The sequence $\binom{n}{k} \mod 2$ for $k = 0$ to $7$ corresponds to the binary digits of $n$ restricted to the first 3 digits (since the binomial coefficient of any bit higher than $k$ would cancel out):

   • For $k = 0$, $\binom{n}{0} = 1$.
   • For $k = 1$, $\binom{n}{1} = 1$ (if the least significant bit is 1).
   • Continue for all $k = 0$ to $7$.

The sequence is therefore: $\text{Sequence} = 11000000$

This sequence comes from the fact that $n$ has 1s in the first and second least significant positions.

Summary:

• Question 3: The binary result was an attempt based on a function $g(n)$ with the hypothesis that could be incorrect as it does not yield standard expected results.
• Question 4: The correct answer sequence is 11000000.

Would you like to go through the steps in more detail or explore further examples?

---

Relative Questions:

1. How can Lucas' theorem be generalized for other moduli?
2. What are the properties of binary representations when considering bit manipulation?
3. What would $g(n)$ be if we modified the problem to use different weights?
4. What happens if $w(n)$ is a very large number, like in the case of very dense binary numbers?
5. How does one efficiently compute large binomial coefficients modulo small primes?

Tip: Always try decomposing complex binary or modular problems using smaller, simpler examples to verify your hypotheses.