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 A061057 Factorial splitting: write n! = x*y with x <= y and x maximal; sequence gives value of y-x. 6
 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 (list; graph; refs; listen; history; text; internal format)
 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 Robert G. Wilson v, Table of n, a(n) for n = 1..41 (first 40 terms from Max Alekseyev) 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 CROSSREFS Cf. A061055, A061056, A061058, A061059, A061060. Cf. A061030, A061031, A061032, A061033. Cf. A005563, A038507. Cf. A038667. - Robert G. Wilson v, Jul 12 2009 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 STATUS approved

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Last modified April 13 21:24 EDT 2021. Contains 342941 sequences. (Running on oeis4.)