Values \(m = \sigma(k)\) ($0 < m… to Number of integers $0 < k \le…
- Values \(m = \sigma(k)\) (\(0 < m < 2n\)) with \(2n - 1 - m\) and \(2n - 1 + m\) prime
- Number of integers \(0 < k < n\) such that \(k \phi(n - k) + 1\) is a square
- Number of integers \(0 < k < n\) such that \((k + 1)^{\phi(n - k)} + k\) is prime
- Smallest \(x\) such that \(x \bmod \phi(x) = n\), or \(0\) if no such \(x\) exists
- Integers \(0 < k < n\) with \(p = k + \mathrm{prime}(n - k)\) and \(\mathrm{prime}(p) - p + 1\) prime
- Number of integers \(0 < k < n\) such that \(p = k + \phi(n - k)\) and \(2(n - p) + 1\) are both prime
- Number of \(0 < k < n-2\) with \(p = \phi(k) + \phi(n-k)/2 + 1\) and \(p_p - p \pm 1\) prime
- Prime quadruplets generated by \(p = 3\phi(k) + \phi(n-k) - 1\)
- Ordered ways to write \(2n = p + q\) with \(p, q\) and \(p_{p+2} + 2\) all prime
- Number of \(0 < k < n/2\) such that \(\phi(k)\phi(n-k)\) is a square
- Number of parts in the symmetric representation of \(\sigma(n)\)
- Ordered ways to write \(n = k + m\) with \(p_k + 4\) and \(p_{p_m} + 4\) both prime
- Ways to write \(n = k + m\) with \(p_k^2 - 2\), \(p_m^2 - 2\), and \(p_{p_m}^2 - 2\) all prime
- Number of \(0 < k < n\) such that \(\pi(kn)\) is prime
- Number of primes \(p \le (n+1)/2\) with \(\lfloor\sqrt{n-p}\rfloor\) prime
- Pairs \(\{j, k\}\) with \(0 < j < k \le n\), \(k \equiv 1 \pmod j\), and \(\pi(jn) \mid \pi(kn)\)
- Equal prime counts in adjacent intervals \((kn, (k+1)n)\) and \(((k+1)n, (k+2)n)\)
- Number of positive integers \(k < n\) such that \(n^2 - \pi(k n)\) is prime
- Number of primes \(p < n\) with \(\text{prime}(p)^2 + (\text{prime}(n) - 1)^2\) prime
- Number of integers \(0 < k \le n\) such that \(\pi(\pi(k n))\) is a square
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