A375077 Smallest k such that Product_{i=0..n} (k-i) divides C(2k,k).

2, 2480, 8178, 45153, 3648841, 7979090, 101130029

Offset: 1

Author

Ralf Stephan, Jul 29 2024

Keywords

• nonn
• hard
• more
• changed

Links

• Thomas Bloom, Problem 396, Erdős Problems.

Crossrefs

Cf. A000984, A120626, A129489.

Programs

PARI

for(n=1,20,for(k=n+1,100000,if(binomial(2*k,k)%prod(i=0,n,k-i)==0,print(n," ",k);break)))

Python

from math import prod, comb
def A375077(n):
    a, c, k = prod(n+1-i for i in range(n+1)), comb(n+1<<1,n+1), n+1
    while c%a:
        k += 1
        a = a*k//(k-n-1)
        c = c*((k<<1)-1<<1)//k
    return k # Chai Wah Wu, Jul 30 2024

Extensions

a(5) from Chai Wah Wu, Aug 01 2024

a(6)-a(7) from Max Alekseyev, Feb 25 2025