According to Eddington, the exact number of protons in the universe, where 136 was the reciprocal of the fine structure constant as best as it could be measured in his time.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 15 and 49, 1999.

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Weisstein, Eric W. "Eddington Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EddingtonNumber.html