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

1,4

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

Links

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

Formula

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

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

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A036655 A319356 A098792 * A038667 A304104 A199823

Adjacent sequences: A061054 A061055 A061056 * A061058 A061059 A061060

Keyword

nonn

Author

Ed Pegg Jr, May 28 2001

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

Status

approved