Schulte’s Exponent-Product Multiplicativity Conjecture

Abstract

Schulte’s A322327 exponent-product function is multiplicative with prescribed prime powers.

N denotes the natural numbers including zero and Z denotes the integers. For n in N, primeFactors(n) is the finite set of prime divisors, factorization(n,p), also written v_p(n), is the natural exponent of p in n, and A005361(n) is the product of those exponents. The empty product gives A005361(1)=1. The function omega(n) counts distinct prime divisors, k is an arbitrary integer parameter, and a_k(n) is integer valued. Coprimality means gcd(m,n)=1. Lean’s power convention gives 0^0=1. The formal claim covers, for every integer k, multiplicativity on coprime natural arguments and the value k times e at every positive prime-power exponent. The trailing OEIS sequence correspondences are outside the claim.

Definition 1.1 (The exponent-product function).

$$\forall k\in \mathbb{Z},\forall n\in \mathbb{N},\left(a_k\right)\left(n\right)=\prod _{p\in \mathrm{primeFactors}\left(n\right)}\mathrm{intCast}\left(\mathrm{factorization}\left(n,p\right)\right)\cdot k^{\mathrm{omega}\left(n\right)}$$

Formalization. D5/S3/Arith/SchulteExponentProductOmegaPowerMultiplicative.a (✓ std3).

Citation. Werner Schulte (2018). OEIS A322327, A005361(n)·A034444(n), with the conjecture that A005361(n)·k^ω(n) is multiplicative with prime-power values k·e. URL: https://oeis.org/A322327.

Commentary.

The finite product is A005361(n); each natural exponent is explicitly cast to an integer before multiplication by the integer power k^omega(n).

Theorem 1.2 (Schulte’s multiplicativity and prime-power formula).

$$\forall k\in \mathbb{Z},(\forall m\in \mathbb{N},\forall n\in \mathbb{N},(\gcd \left(m,n\right)=1)\Rightarrow (\left(a_k\right)\left(m\cdot n\right)=\left(a_k\right)\left(m\right)\cdot \left(a_k\right)\left(n\right)))\wedge (\forall p\in \mathbb{N},\forall e\in \mathbb{N},(\mathrm{Prime}\left(p\right))\Rightarrow ((0<e)\Rightarrow (\left(a_k\right)\left(p^e\right)=k\cdot \mathrm{intCast}\left(e\right))))$$

Proof. Machine-checked in Lean as D5/S3/Arith/SchulteExponentProductOmegaPowerMultiplicative.result (✓ std3). ∎

Resolves. Problems/oeis-a322327-schulte-exponent-product-omega-power-multiplicative (proved) by D5/S3/Arith/SchulteExponentProductOmegaPowerMultiplicative.result.

Citation. Werner Schulte (2018). OEIS A322327, A005361(n)·A034444(n), with the conjecture that A005361(n)·k^ω(n) is multiplicative with prime-power values k·e. URL: https://oeis.org/A322327.

Commentary.

For every integer k, the first conjunct states multiplicativity on coprime natural arguments and the second gives a_k(p^e)=k times e for prime p and positive e. This settles the two assertions in the OEIS conjecture sentence within the stated scope.

References

• Truth anchor: D5/S3/Arith/SchulteExponentProductOmegaPowerMultiplicative.a
• Truth anchor: D5/S3/Arith/SchulteExponentProductOmegaPowerMultiplicative.result