Maximum Number of Divisors

Let $d(n)$ be the number of divisors of $n$.
Let $M(n,k)$ be the maximum value of $d(j)$ for $n \le j \le n+k-1$.
Let $S(u,k)$ be the sum of $M(n,k)$ for $1 \le n \le u-k+1$.

You are given that $S(1000,10)=17176$.

Find $S(100\,000\,000, 100\,000)$.

To find the value of f(10^14), we need to count the number of proper squarefree Gaussian integers with a^2 + b^2 ≤ 10^14.

To do this, we can iterate through all possible values of a and b, where a^2 + b^2 ≤ 10^14. For each pair (a, b) that satisfies the condition, we check if the Gaussian integer z = a + bi is squarefree by checking its prime factorization. If z is squarefree, we increment the count.

We also need to check if z is proper, i.e., a > 0 and b ≥ 0. If a = 0 or b < 0, we skip to the next iteration. To check if a Gaussian integer is squarefree, we need to factorize it into its prime factors. We can use a modified version of the Sieve of Eratosthenes to find all the Gaussian primes less than or equal to the given limit. Here's a Python code that solves the problem: ```python import math def count_squarefree(limit): primes = get_gaussian_primes(limit) # Get all Gaussian primes up to the limit count = 0 for a in range(1, int(math.sqrt(limit)) + 1): for b in range(int(math.sqrt(limit - a*a)) + 1): z = a + b * 1j # Create Gaussian integer if is_proper(z): if is_squarefree(z, primes): count += 1 return count def is_proper(z): return z.real > 0 and z.imag >= 0

def is_squarefree(z, primes):
for prime in primes:
if prime == z:
continue
if prime.real * prime.real + prime.imag * prime.imag == 1:
if z.real % prime.real == 0 and z.imag % prime.imag == 0:
return False
else:
if z.real % prime.real == 0 and z.imag % prime.imag == 0 and z.real == prime.real and z.imag == prime.imag:
return False

return True

def get_gaussian_primes(limit):
primes = []
sieve = [True] * (limit + 1)
sieve[0] = False
sieve[1] = False

for p in range(2, int(math.sqrt(limit)) + 1):
if sieve[p]:
for i in range(p*p, limit + 1, p):
sieve[i] = False

for a in range(0, int(math.sqrt(limit)) + 1):
for b in range(a + 1):
if a == 0 and b == 0:
continue
z = a + b * 1j
if sieve[a*a + b*b]:
primes.append(z)

return primes

result = count_squarefree(10**14)
print(result)
“`

The code uses the `count_squarefree` function to calculate the count of proper squarefree Gaussian integers with a^2 + b^2 ≤ 10^14.
The `get_gaussian_primes` function implements the modified Sieve of Eratosthenes to get all the Gaussian primes up to the given limit.
The `is_proper` function checks whether a Gaussian integer is proper.
The `is_squarefree` function checks whether a Gaussian integer is squarefree by iterating through the Gaussian primes and checking if any of them divide the given Gaussian integer.

Running this code will give the value of $f(10^{14})$.

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