Odd and even k-full and l-free numbers

TheoremOdd-to-even ratio of \(k\)-full numbers#●ready to prove

Let \(k \ge 2\). For a natural number \(N\) let \(K_o(N)\) and \(K_e(N)\) be the numbers of odd and of even \(k\)-full numbers \(n \le N\). Then \(K_o(N)/K_e(N) \to 2 - 2^{1 - 1/k}\) as \(N \to \infty\).

The counts up to a real \(x\) depend only on \(\lfloor x \rfloor\), so this is the same as the limit as \(x \to \infty\) through the reals.

public theorem_wanted tendsto_card_odd_kFull_div_card_even_kFull {k : ℕ} (hk : 2 ≤ k) :
    Tendsto (fun N : ℕ =>
        ({n : ℕ | n ≤ N ∧ Nat.IsKFull k n ∧ Odd n}.ncard : ℝ) /
          {n : ℕ | n ≤ N ∧ Nat.IsKFull k n ∧ Even n}.ncard)
      atTop (𝓝 (2 - (2 : ℝ) ^ (1 - 1 / (k : ℝ))))

Sources

• Sunanta Srisopha and Teerapat Srichan, The Odd/Even Dichotomy for the Set of Numbers That Are Both k-Full and l-Free, INTEGERS 24 (2024), Article A106, Corollary 2.

Uses 1

Statement uses\(k\)-full numbers

TheoremOdd-to-even ratio of \(k\)-full, \(l\)-free numbers#●ready to prove

Let \(2 \le k < l\). Call a natural number \(n\) a \([k, l]\)-integer if it is \(k\)-full and \(l\)-free, that is, \(n \ge 1\) and every prime \(p\) dividing \(n\) satisfies \(p^k \mid n\) and \(p^l \nmid n\). For a natural number \(N\) let \(O_{k,l}(N)\) and \(E_{k,l}(N)\) be the numbers of odd and of even \([k, l]\)-integers \(n \le N\). Then \(O_{k,l}(N)/E_{k,l}(N) \to (2 - 2^{1 - 1/k})/(1 - 2^{1 - l/k})\) as \(N \to \infty\).

The counts up to a real \(x\) depend only on \(\lfloor x \rfloor\), so this is the same as the limit as \(x \to \infty\) through the reals.

public theorem_wanted tendsto_card_odd_kFull_lFree_div_card_even_kFull_lFree {k l : ℕ}
    (hk : 2 ≤ k) (hkl : k < l) :
    Tendsto (fun N : ℕ =>
        ({n : ℕ | n ≤ N ∧ Nat.IsKFull k n ∧ n.IsPowerFree l ∧
            Odd n}.ncard : ℝ) /
          {n : ℕ | n ≤ N ∧ Nat.IsKFull k n ∧ n.IsPowerFree l ∧
            Even n}.ncard)
      atTop (𝓝 ((2 - (2 : ℝ) ^ (1 - 1 / (k : ℝ))) / (1 - (2 : ℝ) ^ (1 - (l : ℝ) / k))))

Sources

• Sunanta Srisopha and Teerapat Srichan, The Odd/Even Dichotomy for the Set of Numbers That Are Both k-Full and l-Free, INTEGERS 24 (2024), Article A106, Theorem 1.

Uses 2

Statement uses\(k\)-full numbers · Power-free natural numbers

TheoremOdd-to-even ratio of \(l\)-free numbers#●ready to prove

Let \(l \ge 2\). For a natural number \(N\) let \(O(N)\) and \(E(N)\) be the numbers of odd and of even \(l\)-free positive integers \(n \le N\). Then \(O(N)/E(N) \to 2^l/(2^l - 2)\) as \(N \to \infty\).

This is the case \(k = 1\) of the odd-to-even ratio for \(k\)-full, \(l\)-free numbers, since every positive integer is \(1\)-full.

public theorem_wanted tendsto_card_odd_lFree_div_card_even_lFree {l : ℕ} (hl : 2 ≤ l) :
    Tendsto (fun N : ℕ =>
        ({n : ℕ | n ≤ N ∧ Nat.IsKFull 1 n ∧ n.IsPowerFree l ∧
            Odd n}.ncard : ℝ) /
          {n : ℕ | n ≤ N ∧ Nat.IsKFull 1 n ∧ n.IsPowerFree l ∧
            Even n}.ncard)
      atTop (𝓝 ((2 : ℝ) ^ l / ((2 : ℝ) ^ l - 2)))

Sources

• Sunanta Srisopha and Teerapat Srichan, The Odd/Even Dichotomy for the Set of Numbers That Are Both k-Full and l-Free, INTEGERS 24 (2024), Article A106, Corollary 1.

Uses 2

Statement uses\(k\)-full numbers · Power-free natural numbers

3 ready to prove

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