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	OEIS	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

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.