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

1,2

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.

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.

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

Example

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.

Mathematica

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

Crossrefs

Cf. A027423, A055226, A000196, A000142, A000005, A060776, A061057, A033677.

Sequence in context: A261230 A014278 A061056 * A200744 A089430 A337638

Adjacent sequences: A060774 A060775 A060776 * A060778 A060779 A060780

Keyword

nonn

Author

Labos Elemer, Apr 26 2001

Extensions

More terms from Don Reble, Dec 13 2001

Status

approved