A063191
Engel expansion of Sum_{k>=0} 1/(8 + k)^k.
Original entry on oeis.org
1, 9, 11, 15, 25, 27, 36, 173, 202, 1965, 10312, 13538, 31564, 39844, 88358, 90936, 171767, 798975, 1350952, 1934096, 4998490, 6950469, 16097930, 30366857, 201980012, 280003359, 726993911, 6157450525, 25921409405, 26415716358
Offset: 1
Cf.
A006784 for definition of Engel expansion.
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ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A063192
Engel expansion of Sum_{k>=0} 1/(9 + k)^k.
Original entry on oeis.org
1, 10, 12, 16, 21, 22, 33, 64, 64, 372, 7928, 10206, 18400, 319831, 429477, 588460, 849875, 859279, 2903676, 11021989, 14742559, 16613535, 48017191, 329292950, 3988366898, 5315516684, 19567917130, 21634744969, 23775894403
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 ]
A063193
Engel expansion of Sum_{k>=0} 1/(10 + k)^k.
Original entry on oeis.org
1, 11, 13, 17, 19, 22, 39, 46, 95, 100, 202, 716, 1008, 1347, 3544, 4174, 27864, 51834, 81170, 115510, 172502, 1297395, 1428481, 3429593, 8484143, 16796993, 35875141, 202879810, 560789726, 953079326, 1456311045, 2218799342
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 ]
A063194
Engel expansion of 1 + Sum_{k>=1} 1/k^k.
Original entry on oeis.org
1, 1, 4, 7, 7, 11, 80, 114, 6099, 11431, 26617, 33585, 56088, 61344, 194790, 234794, 1389968, 1786243, 2418760, 2647328, 3746318, 4952495, 5249786, 5341636, 10517386, 55449405, 79637649, 561022295, 625320951, 3650616781
Offset: 1
-
ToEngel[ x_, n_Integer ] := Rest@First@Transpose@NestList[ {Ceiling[ 1/# ], #}&[ Times@@#-1 ]&, {1, Abs[ x ]+1}, n ]
A067914
Engel expansion of zeta(6)=sum(i>0,1/i^6).
Original entry on oeis.org
1, 58, 170, 386, 5254, 66282, 599652, 868113, 3766902, 103572028, 168604143, 184143987, 435313165, 487337902, 2137572181, 7806336112, 58951218998, 88988779166, 98134403047, 151936444428, 445419603341, 712997180935
Offset: 1
See
A006784 for explanation of Engel expansions.
A067915
Engel expansion of zeta(7)=sum(i>0,1/i^7).
Original entry on oeis.org
1, 120, 523, 1543, 1615, 9192, 16003, 21589, 101625, 148947, 169047, 362415, 497588, 745709, 815536, 4121913, 59871203, 199990957, 309564563, 361924763, 1029742912, 1571746050, 7713398235, 18093587867, 81964002218, 483653860925
Offset: 1
See
A006784 for explanation of Engel expansions.
A067916
Engel expansion of zeta(8)=sum(i>0,1/i^8).
Original entry on oeis.org
1, 246, 331, 357, 602, 13543, 13977, 41685, 42358, 81233, 1797406, 3474721, 3635552, 12183608, 79438338, 101944448, 874415778, 1751624804, 3935310958, 4054291158, 8855245981, 16925204733, 70558648332, 168243185799
Offset: 1
See
A006784 for explanation of Engel expansions.
A067917
Engel expansion of zeta(9)=sum(i>0,1/i^9).
Original entry on oeis.org
1, 498, 5568, 6050, 13355, 19997, 86532, 1232762, 55372414, 58828570, 515325171, 585021906, 3531704172, 8467462253, 33628341236, 39825271855, 103454395889, 619628068650, 864282158571, 2124680840763, 4385336400732, 7477492931412
Offset: 1
See
A006784 for explanation of Engel expansions.
A067918
Engel expansion of zeta(10) = Sum_{i>0} 1/i^10.
Original entry on oeis.org
1, 1006, 1844, 1943, 2121, 6730, 9457, 16986, 26554, 51607, 190310, 624191, 982911, 8532607, 228180184, 328852129, 1087944970, 3446300146, 6297250622, 13963928263, 21739950459, 22065516615, 40578950043, 147724913629, 979260576959, 988238658616, 3024618853544
Offset: 1
See
A006784 for explanation of Engel expansions.
-
\\ a(1)=1 and with 1500 significant digits:
s=zeta(10); for(i=1,30,s=s*ceil(1/s)-1; print1(ceil(1/s),","); );
A067920
Engel expansion of sin(2).
Original entry on oeis.org
2, 2, 2, 4, 11, 14, 57, 100, 1778, 2355, 7690, 21334, 89831, 126718, 365351, 665175, 914152, 1469797, 3554254, 25042522, 190651419, 580715831, 1803513148, 3705699670, 44927205487, 232354919706, 1600668490436, 7538452976365, 12294187702836, 28817421277388
Offset: 1
See
A006784 for explanation of Engel expansions.
-
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]]; EngelExp[N[Sin[2], 7!], 30] (* Georg Fischer, Nov 20 2020 *)
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my(r=sin(2)); for(i=1, 20, my(s=r*ceil(1/r)-1); print1(ceil(1/r), ", "); r=s); /* Benoit Cloitre [amended by Georg Fischer, Nov 20 2020] */
Comments