E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant given by
(1)
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which is good to 6 digits.
M. Hudson (pers. comm., Nov. 8, 2004) found the approximations
(2)
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(3)
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(4)
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(5)
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(6)
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(7)
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where is the Euler-Mascheroni constant and is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.
A curious approximation to is given by
(8)
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where is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).
Lima (2009) found the approximation
(9)
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where is Catalan's constant, which is correct to 21 digits.