A054544
Engel series expansion (or "Egyptian product") for Khintchine's constant.
Original entry on oeis.org
1, 1, 2, 3, 9, 70, 117, 503, 648, 1078, 12868, 41235, 178650, 377670, 394301, 546185, 2600672, 8729780, 41318679, 83367169, 525961060, 561571346, 1556964264, 1868773845, 15139200289, 27297789005, 30324107039, 56699922000
Offset: 0
- Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 59-65.
A059188
Engel expansion of Gamma(1/3) = 2.6789385....
Original entry on oeis.org
1, 1, 2, 3, 14, 33, 57, 236, 6280, 7170, 172302, 24568434, 32871132, 43231756, 60680523, 83128444, 720494727, 803406064, 1804216488, 6655647717, 9106036988, 14962799365, 37839117297, 121819278396, 262108609568
Offset: 1
- F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
-
EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]
A059189
Engel expansion of Gamma(2/3) = 1.35412.
Original entry on oeis.org
1, 3, 17, 17, 50, 79, 796, 3687, 7074, 9098, 95915, 118514, 133188, 186305, 209314, 666015, 5735240, 7685979, 11174747, 97173279, 269061009, 569125952, 932655002, 7282946876, 9919537325, 52110120678, 70254144261
Offset: 1
- F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
-
EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]
A063184
Engel expansion of Sum_{k>=0} 1/(1 + k)^k.
Original entry on oeis.org
1, 2, 4, 36, 2330, 3274, 7195, 28106, 38508, 74032, 74935, 203763, 264407, 431066, 1679581, 2473728, 2790505, 9118184, 10686256, 39154390, 64918230, 3725272877, 6503732261, 10906782833, 33507112881, 34193678041
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063185
Engel expansion of Sum_{k>=0} 1/(2 + k)^k.
Original entry on oeis.org
1, 3, 5, 15, 20, 124, 141, 1085, 1221, 5267, 9814, 9899, 179888, 531293, 43914936, 59108249, 129482155, 290253117, 297264974, 329981693, 515700524, 1791622856, 29237782307, 289017844013, 2187297805011, 20282473409970
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063186
Engel expansion of Sum_{k>=0} 1/(3 + k)^k.
Original entry on oeis.org
1, 4, 6, 13, 16, 31, 131, 231, 578, 905, 1649, 4810, 5999, 10249, 11962, 21338, 229345, 404066, 772217, 1646462, 2129025, 235494066, 559730937, 1287457726, 1549953698, 2919657850, 23892156516, 52085560558, 70347157214
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063187
Engel expansion of Sum_{k>=0} 1/(4 + k)^k.
Original entry on oeis.org
1, 5, 7, 12, 800, 1502, 3021, 5079, 6325, 17537, 50021, 158855, 826545, 5023575, 5097327, 6369490, 17535161, 19462254, 22535278, 24898987, 55666978, 111142756, 305716101, 2653533055, 3591521863, 4889613852, 10577043282
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063188
Engel expansion of Sum_{k>=0} 1/(5 + k)^k.
Original entry on oeis.org
1, 6, 8, 13, 19, 21, 105, 401, 1053, 2012, 4310, 114743, 235156, 287799, 1269345, 1481217, 1978865, 2703836, 3103386, 3325474, 18167905, 553255504, 782224772, 1039603284, 26121336847, 37996375306, 60661762577, 109713190320
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063189
Engel expansion of Sum_{k>=0} 1/(6 + k)^k.
Original entry on oeis.org
1, 7, 9, 13, 132, 259, 756, 995, 1217, 1790, 3809, 6320, 17298, 63963, 266138, 821069, 3300319, 4218118, 4403814, 7172972, 20058471, 92458049, 191069103, 765926110, 822857354, 2489215358, 2980617344, 5257976545, 5926840761, 11328877119, 45478158116, 66167647170
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063190
Engel expansion of Sum_{k>=0} 1/(7 + k)^k.
Original entry on oeis.org
1, 8, 10, 14, 36, 45, 88, 102, 114, 432, 4369, 9183, 15670, 31307, 103514, 511249, 1673274, 2394998, 3225410, 4732293, 54612351, 55335052, 220482735, 39208626697, 290846809341, 296349507792, 2280942643369, 2891972596111
Offset: 1
Cf.
A006784 for definition of Engel expansion.
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
Comments