A060777 Larger central (or median) divisor of n!.

1, 2, 3, 6, 12, 30, 72, 210, 630, 1920, 6336, 22176, 78975, 295680, 1144000, 4576000, 18869760, 80061696, 348986880, 1560176640, 7148445696, 33530112000, 160813154304, 787718131200, 3938590656000, 20083261440000, 104351051284480, 552173794099200, 2973519499493376, 16286922357866496, 90680032493568000, 512971179263262720

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 y. Inequality "x < y" gives the same sequence, except that a(1) is not defined.

The integer part of square root of n! (A055226(n)) is situated between x and y.

Examples

			Divisors of 6!=720 are {1, 2, 3, 4, 5, 6, ..., 24, 30, ..., 360, 720}. a(6)=30, the 16th one from the 30 divisors of 720.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 1..140
• 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, A060776, A061057, A033677.

Programs

Mathematica

Table[ Part[ Divisors[ w! ], 1+Floor[ DivisorSigma[ 0, n! ]/2 ] ], {w, a, b} ]

Formula

a(n) = A033677(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 Don Reble, Dec 13 2001