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

0,2

Comments

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

Links

Table of n, a(n) for n=0..18.

Thomas Bloom, Problem 389.

Example

3*4*5*6 ∣ 7*8*9*10, so a(2) = 4.

Prog

(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

Crossrefs

Sequence in context: A186639 A266668 A043299 * A144776 A371264 A065937

Adjacent sequences: A375068 A375069 A375070 * A375072 A375073 A375074

Keyword

nonn,hard,more

Author

Ralf Stephan, Jul 29 2024

Extensions

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

Status

approved