Arithmetic functions

Divisor sums, totients, Möbius and multiplicative functions, their mean values and extremal orders, and integers classified by them.

• Perfect numbers and their relatives
• Divisor and totient functions
• Multiplicative functions and mean values
• Integers with restricted factorization
• Arithmetic derivative

TheoremThe LCM product identity#✓fully proved

For every natural number \(n\), the product of \(p^{\lfloor n/(p-1) \rfloor}\) over the primes \(p \le n + 1\) equals the least common multiple of all products \(i_1 i_2 \cdots i_n\) of \(n\) positive integers with \(i_1 + \cdots + i_n \le 2n\).

Since \(\lfloor n/(p-1) \rfloor = 0\) for \(p > n + 1\), the left side is the product of \(p^{\lfloor n/(p-1) \rfloor}\) over all primes \(p\).

theorem lcm_product_identity
    (n : ℕ) :
    ∏ p ∈ Nat.primesBelow (n + 2), p ^ (n / (p - 1)) =
      Finset.lcm (Finset.image (fun t : Fin n → Fin (2 * n + 1) => ∏ j, (t j).val)
        (Finset.univ.filter
          (fun t => (∀ j, 1 ≤ (t j).val) ∧ ∑ j, (t j).val ≤ 2 * n))) id

Sources

• Abdelmalek Bedhouche and Bakir Farhi, "On Some Products Taken over Prime Numbers," Journal of Integer Sequences 26 (2023), Article 23.1.5, Corollary, lines 335–340, https://cs.uwaterloo.ca/journals/JIS/VOL26/Farhi/farhi32.tex

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