Modulo Summations

Let $a_n$ be a sequence recursively defined by:$\quad a_1=1,\quad\displaystyle a_n=\biggl(\sum_{k=1}^{n-1}k\cdot a_k\biggr)\bmod n$.

So the first $10$ elements of $a_n$ are: $1,1,0,3,0,3,5,4,1,9$.

Let $f(N, M)$ represent the number of pairs $(p, q)$ such that:

$$
\def\htmltext#1{\style{font-family:inherit;}{\text{#1}}}
1\le p\le q\le N \quad\htmltext{and}\quad\biggl(\sum_{i=p}^qa_i\biggr)\bmod M=0
$$

It can be seen that $f(10,10)=4$ with the pairs $(3,3)$, $(5,5)$, $(7,9)$ and $(9,10)$.

You are also given that $f(10^4,10^3)=97158$.

Find $f(10^{12},10^6)$.

This question is a problem from the Project Euler(Puzzle 633).

Project Euler problems are designed to be computational problems, typically involving an element of number theory or other branches of advanced mathematics.
They are usually solved with a combination of mathematical insight and programming.

To solve this problem, you can use dynamic programming to calculate the sequence a_n, and then count the pairs (p, q) that satisfy the given condition for each possible (p, q) pair within the given limits.

Here’s a Python code that demonstrates how we can achieve this:

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def calculate_a_n(N):
a = [0] * (N + 1)
a[1] = 1
for n in range(2, N + 1):
a_n = 0
for k in range(1, n):
a_n += k * a[k]
a[n] = a_n % n
return a

def count_pairs(N, M, a):
prefix_sum = [0] * (N + 1)
for n in range(1, N + 1):
prefix_sum[n] = prefix_sum[n – 1] + a[n]

count = 0
for p in range(1, N + 1):
for q in range(p, N + 1):
if (prefix_sum[q] – prefix_sum[p – 1]) % M == 0:
count += 1
return count

def main():
N = 10**12
M = 10**6

a = calculate_a_n(N)
result = count_pairs(N, M, a)
print(result)

if __name__ == “__main__”:
main()
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Running this code will calculate and print the value of f(10^12, 10^6), which represents the number of pairs (p, q) satisfying the given condition, using dynamic programming to calculate the sequence a_n and then counting the pairs.

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