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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.

%I A375071 #44 Sep 01 2025 11:23:39
%S A375071 1,5,4,207,206,2475,984,8171,8170,45144,45143,3648830,3648829,7979077,
%T A375071 7979076,58068862,58068861,255278295,255278294
%N 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.
%C A375071 Erdős and Straus asked if a(n) exists for all n.
%H A375071 Thomas Bloom, <a href="https://www.erdosproblems.com/389">Problem 389</a>, Erdős Problems.
%e A375071 3*4*5*6 ∣ 7*8*9*10, so a(2) = 4.
%o A375071 (PARI) a(n) = my(k=1); while(prod(i=1, k, n+k+i)%prod(i=1, k, n+i), k++); k;
%o A375071 (Python)
%o A375071 from itertools import count
%o A375071 def A375071(n):
%o A375071     a, b = n+1, n+2
%o A375071     for k in count(1):
%o A375071         if not b%a:
%o A375071             return k
%o A375071         a *= n+k+1
%o A375071         b = b*(n+2*k+1)*(n+2*k+2)//(n+k+1) # _Chai Wah Wu_, Aug 01 2024
%K A375071 nonn,hard,more,changed
%O A375071 0,2
%A A375071 _Ralf Stephan_, Jul 29 2024
%E A375071 a(10)-a(18) from _Bhavik Mehta_, Aug 16 2024