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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 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

Continued Fraction, Egyptian Fraction, Pierce Expansion

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## 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.

Engel Expansion

## Cite this as:

Weisstein, Eric W. "Engel Expansion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EngelExpansion.html