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A094331
Least k such that n! < (n+1)(n+2)(n+3)...(n+k).
4
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
1,3
LINKS
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.
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
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 *)
PROG
(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
CROSSREFS
Cf. A075357.
Sequence in context: A121930 A020909 A075357 * A135672 A265133 A064506
KEYWORD
nonn
AUTHOR
Amarnath Murthy, May 15 2004
EXTENSIONS
Corrected and extended by Ray Chandler, May 23 2004
STATUS
approved