A138196 Number of different ways n! can be represented as the difference of two squares; also, for n >= 4, half the number of positive integer divisors of n!/4.

1, 0, 0, 2, 4, 9, 18, 36, 60, 105, 210, 324, 648, 1080, 1680, 2352, 4704, 6480, 12960, 18360, 27200, 43200, 86400, 110880, 155232, 243936, 310464, 423360, 846720, 1080000, 2160000, 2592000, 3686400, 5713920, 7713792, 9237888, 18475776

Offset: 1

Author

John T. Robinson (jrobinson(AT)acm.org), May 04 2008

Keywords

• nonn

Comments

For maximal value x such that x^2 - y^2 = n! see A139151, for maximal value y such that x^2 - y^2 = n! see A181892. - Artur Jasinski, Mar 31 2012

Examples

			a(5)=4 since 5! = 120 = 31^2 - 29^2 = 17^2 - 13^2 = 13^2 - 7^2 = 11^2 - 1^2.
		

Links

• Sudipta Mallick, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A139151, A181892.

Programs

Maple

A138196 := proc(n)
        if n <= 3 then
                op(n,[1,0,0]) ;
        else
                numtheory[tau](n!/4)/2 ;
        end if;
end proc: # R. J. Mathar, Apr 03 2012

Mathematica

(* for n>=4 *) cc = {}; Do[w = n!/4; kk = Floor[(DivisorSigma[0, w] + 1)/2]; AppendTo[cc, kk], {n, 4, 54}]; cc (* Artur Jasinski, Mar 31 2012 *)

PARI

a(n) = if (n<4, (n==1), numdiv(n!/4)/2); \\ Michel Marcus, Jun 22 2019

Formula

For n >= 4, if p_i is the i-th prime, with p_k the largest prime not exceeding n and n!/4 = (p_1^e_1)*(p_2^e_2)* ... *(p_k^e_k), then a(n) = (1/2)*(e_1+1)*(e_2_+1)* ... *(e_k+1).