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A138196
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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.
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3
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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FORMULA
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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).
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EXAMPLE
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a(5)=4 since 5! = 120 = 31^2 - 29^2 = 17^2 - 13^2 = 13^2 - 7^2 = 11^2 - 1^2.
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MAPLE
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if n <= 3 then
op(n, [1, 0, 0]) ;
else
numtheory[tau](n!/4)/2 ;
end if;
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MATHEMATICA
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(* 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 *)
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PROG
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(PARI) a(n) = if (n<4, (n==1), numdiv(n!/4)/2); \\ Michel Marcus, Jun 22 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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John T. Robinson (jrobinson(AT)acm.org), May 04 2008
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STATUS
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approved
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