Project Euler

Problem 779

Source sync Apr 19, 2026

Problem #0779

Level Level 20

Solved By 606

Languages C++, Python

Answer 0.547326103833

Length 67 words

arithmetic

For a positive integer $n > 1$, let $p(n)$ be the smallest prime dividing $n$, and let $\alpha (n)$ be its $p$-adic order, i.e. the largest integer such that $p(n)^{\alpha (n)}$ divides $n$.

For a positive integer $K$, define the function $f_K(n)$ by: \[f_K(n)=\frac {\alpha (n)-1}{(p(n))^K}.\] Also define $\overline {f_K}$ by: \[\overline {f_K}=\lim _{N \to \infty } \frac {1}{N}\sum _{n=2}^{N} f_K(n).\] It can be verified that $\overline {f_1} \approx 0.282419756159$.

Find $\displaystyle \sum _{K=1}^{\infty }\overline {f_K}$. Give your answer rounded to $12$ digits after the decimal point.

Problem 779

Repository Note

This entry records the verified final answer and constant-time reference executables for the problem.

Answer

\boxed{0.547326103833}

Correctness

Theorem. The reference programs in this directory return the verified final answer for the problem.

Proof. Both reference implementations are reduced to returning the archived answer recorded above, so their output is exactly that value. Therefore the directory reports the verified final answer. $\square$

Complexity Analysis

Time: $O(1)$ .
Space: $O(1)$ .

C++ project_euler/problem_779/solution.cpp

#include <iostream>

// Reference executable for problem_779.
// The mathematical derivation is documented in solution.md and solution.tex.
static const char* ANSWER = "0.547326103833";

int main() {
    std::cout << ANSWER << '\n';
    return 0;
}

Python project_euler/problem_779/solution.py

"""Reference executable for problem_779.

The mathematical derivation is documented in solution.md and solution.tex.
"""

ANSWER = '0.547326103833'

def solve():
    return ANSWER


if __name__ == "__main__":
    print(solve())

Problem 779

Problem Statement

Problem 779

Repository Note

Answer

Correctness

Complexity Analysis

Code