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, 1019547825, 1019547824, 17609764973, 17609764972, 1070858041562, 1070858041561, 5048891644621, 5048891644620

Offset

0,2

Comments

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

It can be shown that this is very similar to A375077, with the transformation (n, k) -> (n+k-1, n), though this doesn't guarantee minimality - a(7) here is much smaller than expected. - Sharvil Kesarwani, Mar 30 2026

Links

Table of n, a(n) for n=0..26.

Thomas Bloom, Problem 389, Erdős Problems.

Erdős problems database contributors, Erdős problem database, see no. 389.

Example

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

Mathematica

Table[k = 1; While[Not[IntegerQ[Binomial[n + 2 k, k]/Binomial[n + k, n]]], k++]; k, {n, 0, 10}] (* Vaclav Kotesovec, Apr 01 2026 *)

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

a(19)-a(27) from Sharvil Kesarwani, Mar 30 2026

Status

approved