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A094331 Least k such that n! < (n+1)(n+2)(n+3)...(n+k).

%I A094331 #13 Jan 05 2025 19:51:37
%S A094331 1,1,2,2,3,4,4,5,6,6,7,8,8,9,10,11,11,12,13,13,14,15,16,16,17,18,18,
%T A094331 19,20,21,21,22,23,24,24,25,26,27,27,28,29,30,30,31,32,33,33,34,35,36,
%U A094331 37,37,38,39,40,40,41,42,43,43,44,45,46,47,47,48,49,50,50,51,52,53,53,54
%N A094331 Least k such that n! < (n+1)(n+2)(n+3)...(n+k).
%H A094331 Michael S. Branicky, <a href="/A094331/b094331.txt">Table of n, a(n) for n = 1..10000</a>
%H A094331 Michael A. Brilleslyper, Nathan Wakefield, A. J. Wallerstein, and Bradley Warner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/54-1/BrilleslyperWakefieldWallersteinWarner.pdf">Comparing the Growth of the Prime Numbers to the Natural Numbers</a>, Fibonacci Quart. 54 (2016), no. 1, 65-71.
%F A094331 a(n) = f(n) + 1, where f(n) is the function defined on p. 65 of Brilleslyper et al. - _Michel Marcus_, Apr 11 2022
%t A094331 lk[n_]:=Module[{k=1,f=n!},While[f>=Times@@Table[n+i,{i,k}],k++];k]; Array[lk,80] (* _Harvey P. Dale_, Sep 20 2016 *)
%o A094331 (PARI) a(n) = my(k=0); while (n!^2 >= (n+k)!, k++); k; \\ _Michel Marcus_, Apr 11 2022
%o A094331 (Python)
%o A094331 from math import factorial
%o A094331 def a(n):
%o A094331     fn, k, p = factorial(n), 1, n+1
%o A094331     while fn >= p: k += 1; p *= (n+k)
%o A094331     return k
%o A094331 print([a(n) for n in range(1, 75)]) # _Michael S. Branicky_, Apr 11 2022
%Y A094331 Cf. A075357.
%K A094331 nonn
%O A094331 1,3
%A A094331 _Amarnath Murthy_, May 15 2004
%E A094331 Corrected and extended by _Ray Chandler_, May 23 2004