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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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

LINKS

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

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).

EXAMPLE

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

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 *)

PROG

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

CROSSREFS

Cf. A139151, A181892.

Sequence in context: A065055 A065030 A103321 * A298404 A101351 A293333

Adjacent sequences:  A138193 A138194 A138195 * A138197 A138198 A138199

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified October 20 10:45 EDT 2019. Contains 328257 sequences. (Running on oeis4.)