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Engel Expansion
The Engel expansion, also called the Egyptian product, of a positive real number  is the unique increasing sequence  of
positive integers  such that
The following table gives the Engel expansions of Catalan's constant, e, the Euler-Mascheroni
constant  ,  , and the golden
ratio  .
| constant | Sloane | Engel expansion |  | A028254 | 1,
3, 5, 5, 16, 18, 78, 102, 120, ... |  | A028257 | 1, 2, 3, 3, 6, 17, 23, 25, 27, 73, ... |  | A118239 | 1, 2, 12, 30, 56, 90, 132, 182, ... |  | A000027 | 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... |  | A059193 | 3, 10, 28, 54, 88, 130, 180, 238, 304, 378, ... |  | A053977 | 2, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, ... |  | A054543 | 2, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, ... |  | A059180 | 2, 3, 7, 9, 104, 510, 1413, 2386, ... |  | A028259 | 1, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, ... |  | A006784 | 1, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ... |  | A014012 | 4, 4, 11, 45, 70, 1111, 4423, 5478, 49340, ... |  | A068377 | 1, 6, 20, 42, 72, 110, 156, 210, ... |  | A118326 | 2, 2, 22, 50, 70, 29091, 49606, 174594, ... |
 has a very regular Engel expansion, namely 1, 1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... (OEIS A000027).
Interestingly, the expansion for the hyperbolic sine  has closed form 
for  , which means the expansion for
the hyperbolic cosine  has the closed
form  for  . Similarly,
the Engel expansion for  is 
for  , which follows from
SEE ALSO: Continued Fraction, Egyptian Fraction, Pierce
Expansion
REFERENCES:
Engel, F. "Entwicklung der Zahlen nach Stammbruechen." Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg. pp. 190-191,
1913.
Erdős, P. and Shallit, J. O. "New Bounds on the Length of Finite Pierce
and Engel Series." Sem. Theor. Nombres Bordeaux 3, 43-53, 1991.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 53-59,
2003.
Schweiger, F. Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford, England: Oxford
University Press, 1995.
Sloane, N. J. A. Sequences A000027/M0472, A006784/M4475, A014012,
A028254, A028257,
A028259, A053977,
A054543, A059180,
A059193, A068377,
A118239, and A118326
in "The On-Line Encyclopedia of Integer Sequences."
Wu, J. "How Many Points Have the Same Engel and Sylvester Expansions?."
J. Number Th. 103, 16-26, 2003.
Referenced on Wolfram|Alpha: Engel Expansion
CITE THIS AS:
Weisstein, Eric W. "Engel Expansion."
From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/EngelExpansion.html
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![e^(-1)=sum_(n=1)^infty[1/((2n)!)-1/((2n+1)!)].](/National_Library/20161222123739im_/https://mathworld.wolfram.com/images/equations/EngelExpansion/NumberedEquation2.gif)
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