A060776 Smaller central (median) divisor of n!.

1, 1, 2, 4, 10, 24, 70, 192, 576, 1890, 6300, 21600, 78848, 294840, 1143072, 4572288, 18849600, 79968000, 348566400, 1559376000, 7147140000, 33522128640, 160758097500, 787652812800, 3938264064000, 20080974513600, 104348440350000, 552160113120000

Offset: 1

Author

Labos Elemer, Apr 26 2001

Keywords

• nonn

Comments

Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of x. Inequality "x < y" gives the same sequence, except that a(1) is not defined.

Between this d and its complementary divisor, the integer part of square root of n! is situated; for n=6: {24,26,30}. - Nathaniel Johnston, Jun 25 2011

Examples

			Divisors of 6!=720 are {1,2,3,4,5,6,...,24,30,...,360,720}. a(6)=24, the 15th divisor from 30 divisors of 720.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..140 (first 59 terms from Oleg Terentyev)
• Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathematique, Vol. 115, No. 129 (2024), pp. 45-76.

Crossrefs

Cf. A027423, A055226, A000196, A000142, A000005, A060777, A033676.

Programs

Mathematica

Table[ Part[ Divisors[ w! ], Floor[ DivisorSigma[ 0, n! ]/2 ] ], {w, a, b} ]

PARI

a(n) = if (n==1, 1, my(d=divisors(n!)); d[#d\2]); \\ Michel Marcus, Sep 16 2018

Formula

a(n) = n!/A060777(n). - David Wasserman, Jun 15 2002

a(n) = A033676(A000142(n)). - Pontus von Brömssen, Jul 15 2023

Sum_{k=1..n} a(k) = sqrt(n!) * (1 + O(1/n^c)), where c < 1 is a positive constant (De Koninck and Verreault, 2024, p. 48, Theorem 2.1). - Amiram Eldar, Dec 10 2024

Extensions

More terms from David Wasserman, Jun 15 2002

a(27)-a(32) from M. F. Hasler, Sep 20 2011