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Largest factorial <= n.
8

%I #36 Aug 06 2024 07:23:11

%S 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,

%T 24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,24,

%U 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

%N Largest factorial <= n.

%D 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.

%D 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.

%H Charles R Greathouse IV, <a href="/A048764/b048764.txt">Table of n, a(n) for n = 1..10000</a>

%H Krassimir T. Atanassov, <a href="http://www.gallup.unm.edu/~smarandache/Atanassov-SomeProblems.pdf">On Some of Smarandache's Problems</a>.

%H Li Jie, <a href="https://citeseerx.ist.psu.edu/pdf/134f67dafd17bab61928c5a02e2e9808a27a1dad">On the inferior and superior factorial part sequences</a>, in Zhang Wenpeng (ed.), Research on Smarandache Problems in Number Theory (collected papers), 2004, pp. 47-48.

%H Florentin Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>.

%F a(n) >> n log log n / log n. - _Charles R Greathouse IV_, Sep 19 2012

%F From _Amiram Eldar_, Aug 02 2022: (Start)

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

%F In particular:

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

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

%t Table[k = 1; While[(k + 1)! <= n, k++]; k!, {n, 80}] (* _Michael De Vlieger_, Aug 30 2016 *)

%o (PARI) a(n)=my(t=1,k=1);while(t<=n,t*=k++);t/k \\ _Charles R Greathouse IV_, Sep 19 2012

%o (Python)

%o from sympy import factorial as f

%o def a(n):

%o k=1

%o while f(k + 1)<=n: k+=1

%o return f(k)

%o print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 21 2017, after Mathematica code

%Y Cf. A000142, A001113, A096789.

%K nonn,easy

%O 1,2

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