A375071 Smallest k such that Product_{i=1..k} (n+i) divides Product_{i=1..k} (n+k+i), or 0 if there is no such k.

1, 5, 4, 207, 206, 2475, 984, 8171, 8170, 45144, 45143, 3648830, 3648829, 7979077, 7979076, 58068862, 58068861, 255278295, 255278294

Offset: 0

Author

Ralf Stephan, Jul 29 2024

Keywords

• nonn
• hard
• more
• changed

Comments

Erdős and Straus asked if a(n) exists for all n.

Examples

			3*4*5*6 ∣ 7*8*9*10, so a(2) = 4.
		

Links

• Thomas Bloom, Problem 389, Erdős Problems.

Programs

PARI

a(n) = my(k=1); while(prod(i=1, k, n+k+i)%prod(i=1, k, n+i), k++); k;

Python

from itertools import count
def A375071(n):
    a, b = n+1, n+2
    for k in count(1):
        if not b%a:
            return k
        a *= n+k+1
        b = b*(n+2*k+1)*(n+2*k+2)//(n+k+1) # Chai Wah Wu, Aug 01 2024

Extensions

a(10)-a(18) from Bhavik Mehta, Aug 16 2024