Engel Expansion
The Engel expansion, also called the Egyptian product, of a positive real number 
is the unique increasing sequence ![<span style=]()
{a_1,a_2,...}" src="https://mathworld.wolfram.com/images/equations/EngelExpansion/Inline2.gif" style="height:15px; width:66px" /> 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.
