A beautiful approximation to the Euler-Mascheroni constant is given by
(1)
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(OEIS A086056; E. W. Weisstein, Apr. 18, 2006), which is good to three decimal digits.
In 1982-1983, Odena gave the strange approximation
(2)
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which is effectively
(3)
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(Munroe 2012).
Castellanos (1988) gave
(4)
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(5)
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(6)
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(7)
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which are good to 6, 8, 9, 14, and 14 digits, respectively.
An approximation involving unit fractions due to P. Galliani (pers. comm., April 1, 2002) is given by
(8)
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which differs from by , i.e., is good to 12 digits.
Ed Pegg, Jr. (pers. comm., March 2, 2002) found
(9)
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which is good to 8 digits.
M. Hudson (pers. comm., Sept. 3, 2004) found the approximations
(10)
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(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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where is the golden ratio, which are good to 5, 5, 6, 7, 7, 8, and 8 digits, respectively.
G. W. Barbosa (pers. comm., Mar. 26 and Apr. 2, 2007) gave
(17)
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(18)
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(19)
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which are good to 10 decimal digits, and where the second approximation is a difference of two pandigital parts. Barbosa (pers. comm., Jan. 7, 2008) also gave the pandigital approximation
(20)
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which is good to 13 decimal digits.