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