A061057 Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y-x.

0, 1, 1, 2, 2, 6, 2, 18, 54, 30, 36, 576, 127, 840, 928, 3712, 20160, 93696, 420480, 800640, 1305696, 7983360, 55056804, 65318400, 326592000, 2286926400, 2610934480, 13680979200, 18906930876, 674165366496, 326850970500, 16753029012720, 16880461678080

Offset: 1

Author

Ed Pegg Jr, May 28 2001

Keywords

• nonn

Comments

Difference between central divisors of n!. - Jaume Oliver Lafont, Mar 13 2009

For n > 1, n! will never be a square, because of primes in the last half of the factors. Therefore the divisors of n! come in pairs (x,y) with x*y = n! and x < y. The sequence gives the difference y-x between the pair nearest to the square root of n!. - Alois P. Heinz, Jul 06 2009

a(n) = 2 iff n belongs to A146968. - Max Alekseyev, Feb 06 2010

Examples

			2! = 1*2, with difference of 1.
3! = 2*3, with difference of 1.
4! = 4*6, with difference of 2.
5! = 10*12, with difference of 2.
6! = 24*30, with difference of 6.
7! = 70*72 with difference of 2.
The corresponding central divisors are two units apart (equivalently, n!+1=A038507(n) is a square) for n = 4, 5, 7 (see A146968).
		

Links

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

Crossrefs

Cf. A000142, A060776, A060777, A060795, A060796, A061060, A061030, A061031, A061032, A061033, A005563, A038507, A038667, A056737, A146968.

Programs

Maple

A060777 := proc(n) local d,nd ; d := sort(convert(numtheory[divisors](n!),list)) ; nd := nops(d) ; op(floor(1+nd/2),d) ; end:
A060776 := proc(n) local d,nd ; d := sort(convert(numtheory[divisors](n!),list)) ; nd := nops(d) ; op(floor(nd/2),d) ; end:
A061057 := proc(n) A060777(n)-A060776(n) ; end:
seq(A061057(n),n=2..27) ; # R. J. Mathar, Mar 14 2009

Mathematica

Do[ With[ {k = Floor[ Sqrt[ x! ] ] - Do[ If[ Mod[ x!, Floor[ Sqrt[ x! ] ] - n ] == 0, Return[ n ] ], {n, 0, 10000000} ]}, Print[ {x, "! =", k, x!/k, x!/k - k} ] ], {x, 3, 22} ]
f[n_] := Block[{k = Floor@ Sqrt[n! ]}, While[ Mod[n!, k] != 0, k-- ]; n!/k - k]; Table[f@n, {n, 2, 32}] (* Robert G. Wilson v, Jul 11 2009 *)
Table[d=Divisors[n!]; len=Length[d]; If[OddQ[len], 0, d[[1 + len/2]] - d[[len/2]]], {n, 34}] (* Vincenzo Librandi, Jan 02 2016 *)

PARI

for(k=2,25,d=divisors(k!);print(d[#d/2+1]-d[#d/2])) \\ Jaume Oliver Lafont, Mar 13 2009

Python

from math import isqrt, factorial
from sympy import divisors
def A061057(n):
    k = factorial(n)
    m = max(d for d in divisors(k,generator=True) if d <= isqrt(k))
    return k//m-m # Chai Wah Wu, Apr 06 2022

Formula

a(n) = A060777(n) - A060776(n).

a(n) = A056737(A000142(n)). - Pontus von Brömssen, Jul 15 2023

Extensions

More terms from Dean Hickerson, Jun 13 2001

Edited by N. J. A. Sloane Jul 07 2009 at the suggestion of R. J. Mathar and Alois P. Heinz

a(41) from Robert G. Wilson v, Oct 03 2014