A159835 Engel expansion of hz = limit_{k -> infinity} 1 + k - Sum_{j = -k..k} exp(-2^j).
1, 4, 4, 4, 4, 6, 11, 11, 11, 14, 61, 266, 1006, 1030, 1261, 6264, 7583, 7979, 7986, 12386, 80041, 87434, 130927, 270073, 1653819, 1715177, 1973657, 3483485, 12346987, 17531499, 21237674, 84103203, 195088616, 725688944, 2813572082, 3138084145, 10870485195
Offset: 1
Examples
hz = 1.3327473824328992250086010983738997044167439822598445365797 ...
References
- F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
Links
- F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
- P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
- Eric Weisstein's World of Mathematics, Engel Expansion
- Wikipedia, Engel Expansion
- Index entries for sequences related to Engel expansions
Programs
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Maple
hz:= limit(1+k -sum(exp(-2^j), j=-k..k), k=infinity): engel:= (r,n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]): Digits:=300: engel(evalf(hz), 39);
Extensions
Some terms corrected by Alois P. Heinz, Nov 22 2020
Comments