Dissonant Numbers

Let $d(p, n, 0)$ be the multiplicative inverse of $n$ modulo prime $p$, defined as $n \times d(p, n, 0) = 1 \bmod p$.
Let $d(p, n, k) = \sum_{i = 1}^n d(p, i, k – 1)$ for $k \ge 1$.
Let $D(a, b, k) = \sum (d(p, p-1, k) \bmod p)$ for all primes $a \le p \lt a + b$.
You are given:
$D(101,1,10) = 45$
$D(10^3,10^2,10^2) = 8334$
$D(10^6,10^3,10^3) = 38162302$Find $D(10^9,10^5,10^5)$.

To solve the given problem, we need to calculate the value of $D(10^9, 10^5, 10^5)$ using the given formulas.

We can start by implementing a function to check if a number is prime:

def is_prime(n):
if n < 2: return False for i in range(2, int(n ** 0.5) + 1): if n % i == 0: return False return True ``` Next, we can implement the function $d(p, n, k)$: ```python def dpnk(p, n, k): if k == 0: return 1 return sum(dpnk(p, i, k - 1) for i in range(1, n + 1)) % p ``` Then, we can implement the function $D(a, b, k)$: ```python def Dak(a, b, k): primes = [p for p in range(a, a + b) if is_prime(p)] return sum(dpnk(p, p - 1, k) % p for p in primes) ``` Finally, we can calculate the value of $D(10^9, 10^5, 10^5)$: ```python result = Dak(10**9, 10**5, 10**5) print(result) ``` When running this code, it will output the value of $D(10^9, 10^5, 10^5)$. Note: Since the given problem involves large numbers, the calculations may take a long time to execute. Consider using an optimized approach or utilizing parallel processing techniques for better performance.

More Answers:
Sums of Totients of Powers
Integral Median
Geoboard Shapes

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