A048764 Largest factorial <= n.

1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24

Offset: 1

Author

Charles T. Le (charlestle(AT)yahoo.com)

Keywords

• nonn
• easy
• changed

References

• J. Castillo, Other Smarandache Type Functions: Inferior/Superior Smarandache f-part of x, Smarandache Notions Journal, Vol. 10, No. 1-2-3 (1999), 202-204.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Krassimir T. Atanassov, On Some of Smarandache's Problems.
• Vassia K. Atanassova and Krassimir T. Atanassov, On the 43rd and 44th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5, No. 2, (1999), 86-88.
• Li Jie, On the inferior and superior factorial part sequences, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 47-48.
• Florentin Smarandache, Only Problems, Not Solutions!.

Crossrefs

Cf. A000142, A001113, A096789, A084558.

Programs

Mathematica

Table[k = 1; While[(k + 1)! <= n, k++]; k!, {n, 80}] (* Michael De Vlieger, Aug 30 2016 *)

PARI

a(n)=my(t=1,k=1);while(t<=n,t*=k++);t/k \\ Charles R Greathouse IV, Sep 19 2012

Python

from sympy import factorial as f
def a(n):
    k=1
    while f(k + 1)<=n: k+=1
    return f(k)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 21 2017, after Mathematica code

Formula

n log log n / log n << a(n) <= n. - Charles R Greathouse IV, Sep 19 2012

From Amiram Eldar, Aug 02 2022: (Start)

Sum_{n>=1} 1/a(n)^m = Sum_{k>=1} k/k!^m (Li Jie, 2004).

In particular:

Sum_{n>=1} 1/a(n)^2 = e (A001113).

Sum_{n>=1} 1/a(n)^3 = BesselI(1,2) (A096789). (End)

a(n) = A000142(A084558(n)). - Ridouane Oudra, Aug 22 2025