2,3

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

Dean Hickerson, Table of n, a(n) for n = 1..36

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 two units apart are:

4!=24 : 4,6.

5!=120 : 10,12.

7!=5040: 70,72.

Equivalently, n!+1 is a square for n=4, n=5 and n=7 (See A038507).

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]

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]

(PARI) for (k=2, 25, d=divisors(k!); n=numdiv(k!); print(d[n/2+1]-d[n/2])) - Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 13 2009

Cf. A061055-A061060, A061030-A061033.

Cf. A005563, A038507.

Cf. A038667. [From Robert G. Wilson v (rgwv(AT)rgwv.com), Jul 12 2009]

Sequence in context: A102628 A036655 A098792 this_sequence A038667 A013608 A130674

Adjacent sequences: A061054 A061055 A061056 this_sequence A061058 A061059 A061060

nonn

Ed Pegg Jr (ed(AT)mathpuzzle.com), May 28 2001

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