Conjecture relating two characterizations of a set of integers.

Informal Statement: For an integer $k ≥ 2$, the following are equivalent:

1. The greatest common divisor of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.

2. Writing prime factorization of k as $k = \prod p_i^{e_i}$, and let $P = \max_i p_i^{e_i}$, one has $k / P > P$.

This conjecture asserts that the integers satisfying (1) are exactly those satisfying (2).

Reference:

• A080170
• A051283

def OeisA080170.GCDCondition (k : ℕ) : Prop

The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$.

Equations

• OeisA080170.GCDCondition k = (((Finset.range k).gcd fun (i : ℕ) => ((i + 2) * k).choose k) = 1)

def OeisA080170.PrimePowerCondition (k : ℕ) : Prop

Let P be the largest prime power dividing k. Then $k / P > P$.

Equations

• OeisA080170.PrimePowerCondition k = (k / Option.getD (Finset.filter IsPrimePow k.divisors).max 0 > Option.getD (Finset.filter IsPrimePow k.divisors).max 0)

theorem OeisA080170.gcdCondition_iff_primePowerCondition (k : ℕ) (hk : 2 ≤ k) : GCDCondition k ↔ PrimePowerCondition k

Conjecture: The gcd condition is equivalent to the prime power condition.