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

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, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 37, 37, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 46, 47, 47, 48, 49, 50, 50, 51, 52, 53, 53, 54

Offset: 1

Author

Amarnath Murthy, May 15 2004

Keywords

• nonn

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Michael A. Brilleslyper, Nathan Wakefield, A. J. Wallerstein, and Bradley Warner, Comparing the Growth of the Prime Numbers to the Natural Numbers, Fibonacci Quart. 54 (2016), no. 1, 65-71.

Crossrefs

Cf. A075357.

Programs

Mathematica

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 *)

PARI

a(n) = my(k=0); while (n!^2 >= (n+k)!, k++); k; \\ Michel Marcus, Apr 11 2022

Python

from math import factorial
def a(n):
    fn, k, p = factorial(n), 1, n+1
    while fn >= p: k += 1; p *= (n+k)
    return k
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Apr 11 2022

Formula

a(n) = f(n) + 1, where f(n) is the function defined on p. 65 of Brilleslyper et al. - Michel Marcus, Apr 11 2022

Extensions

Corrected and extended by Ray Chandler, May 23 2004