A138196 - OEIS

%I #28 Jun 23 2019 00:48:30

%S 1,0,0,2,4,9,18,36,60,105,210,324,648,1080,1680,2352,4704,6480,12960,

%T 18360,27200,43200,86400,110880,155232,243936,310464,423360,846720,

%U 1080000,2160000,2592000,3686400,5713920,7713792,9237888,18475776

%N 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.

%C 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

%H Sudipta Mallick, <a href="/A138196/b138196.txt">Table of n, a(n) for n = 1..1000</a>

%F 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).

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

%p A138196 := proc(n)

%p         if n <= 3 then

%p                 op(n,[1,0,0]) ;

%p         else

%p                 numtheory[tau](n!/4)/2 ;

%p         end if;

%p end proc: # _R. J. Mathar_, Apr 03 2012

%t (* 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 *)

%o (PARI) a(n) = if (n<4, (n==1), numdiv(n!/4)/2); \\ _Michel Marcus_, Jun 22 2019

%Y Cf. A139151, A181892.

%K nonn

%O 1,4

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