|
| |
|
|
A061057
|
|
Factorial splitting: write n! = x*y with x<=y and x maximal; sequence gives value of y-x.
|
|
2
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,4
|
|
|
COMMENTS
| Difference between central divisors of n!. - Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 13 2009
For n>1, n! will never be the square of an integer, 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 Heinz, Jul 06 2009
a(n) = 2 iff n belongs to A146968. [From Max Alekseyev (maxale(AT)gmail.com), Feb 06 2010]
|
|
|
LINKS
| Max Alekseyev, Table of n, a(n) for n = 1..40
|
|
|
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) ; [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 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}] [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 11 2009]
|
|
|
PROG
| (PARI) for(k=2, 25, d=divisors(k!); print(d[#d/2+1]-d[#d/2])) \\ Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 13 2009
|
|
|
CROSSREFS
| Cf. A061055-A061060, A061030-A061033, A005563, A038507.
Cf. A038667. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2009]
Sequence in context: A102628 A036655 A098792 * A038667 A199823 A013608
Adjacent sequences: A061054 A061055 A061056 * A061058 A061059 A061060
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Ed Pegg Jr (ed(AT)mathpuzzle.com), May 28 2001
|
|
|
EXTENSIONS
| More terms from Dean Hickerson (dean.hickerson(AT)yahoo.com), Jun 13, 2001
Edited by N. J. A. Sloane Jul 07 2009 at the suggestion of R. J. Mathar and Alois Heinz
|
| |
|
|
