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Engel Expansion

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The Engel expansion, also called the Egyptian product, of a positive real number x is the unique finite or infinite nondecreasing sequence {a_1,a_2,...} of positive integers a_i such that

x=1/(a_1)+1/(a_1a_2)+1/(a_1a_2a_3)+...,

where an infinite sequence is required not to be eventually constant. Under this convention, the expansion terminates iff x is a rational number (Erdős and Shallit 1991, pp. 43 and 45).

The following table gives the Engel expansions of several constants, including Catalan's constant K, e, the Euler-Mascheroni constant gamma, pi, and the golden ratio phi.

constantOEISEngel expansion
sqrt(2)A0282541, 3, 5, 5, 16, 18, 78, 102, 120, ...
sqrt(3)A0282571, 2, 3, 3, 6, 17, 23, 25, 27, 73, ...
cosh1A1182391, 2, 12, 30, 56, 90, 132, 182, ...
eA0283101, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ...
e^(-1)A0591933, 10, 28, 54, 88, 130, 180, 238, 304, 378, ...
gammaA0539772, 7, 13, 19, 85, 2601, 9602, 46268, 4812284, ...
KA0545432, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, ...
ln2A0591802, 3, 7, 9, 104, 510, 1413, 2386, ...
phiA0282591, 2, 5, 6, 13, 16, 16, 38, 48, 58, 104, ...
piA0067841, 1, 1, 8, 8, 17, 19, 300, 1991, 2492, ...
pi^(-1)A0140124, 4, 11, 45, 70, 1111, 4423, 5478, 49340, ...
sinh1A0683771, 6, 20, 42, 72, 110, 156, 210, ...
tanh1A1183262, 2, 22, 50, 70, 29091, 49606, 174594, ...

e has a very regular Engel expansion, namely 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, ... (OEIS A028310). Interestingly, the expansion for the hyperbolic sine sinh1 has closed form a_n=2(n-1)(2n-1) for n>1, which means the expansion for the hyperbolic cosine cosh1 has the closed form a_n=2(n-1)(2n-3) for n>1. Similarly, the Engel expansion for 1/e is a_n=2(2n+1)(n-1) for n>1, which follows from

e^(-1)=sum_(n=1)^infty[1/((2n)!)-1/((2n+1)!)].

See also

Continued Fraction, Egyptian Fraction, Pierce Expansion

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References

Engel, F. "Entwicklung der Zahlen nach Stammbruechen." In 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. https://doi.org/10.5802/jtnb.41.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 A006784/M4475, A014012, A028254, A028257, A028259, A028310, 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.

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Engel Expansion

Cite this as:

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

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