Program: base-10 "cyclist" number search, by increasing digit-size
Transform: a base-10 positive integer -> raise every one of its digits to the power of the integer's length and sum
Iterate: until a previously encountered integer is reached, thus determining a cycle composed of "cyclists"
Nomenclature: period-1 cyclists are "narcissistic", all other cyclists are "pseudo-altruistic"
More: http://chesswanks.com/blahg/odo/Blog/Entries/2009/2/27_Self-serving_altruistic_numbers.html
Pseudo-Altruistic Numbers (indexed list): http://chesswanks.com/txt/PseudoAltruisticNumbers.txt
Pseudo-Altruistic Numbers (organized by period): http://chesswanks.com/txt/PAN(obp).txt

solution index (cycle length - cycle length index) {cycle}

  1 ( 1- 1)   {1}
  2 ( 1- 2)   {2}
  3 ( 1- 3)   {3}
  4 ( 1- 4)   {4}
  5 ( 1- 5)   {5}
  6 ( 1- 6)   {6}
  7 ( 1- 7)   {7}
  8 ( 1- 8)   {8}
  9 ( 1- 9)   {9}
  
 10 ( 1-10)   {370}
 11 ( 3- 1)   {160, 217, 352}
 
 12 ( 1-11)   {153}
 13 ( 1-12)   {371}
 14 ( 2- 1)   {136, 244}
 15 (10- 1)   {259, 862, 736, 586, 853, 664, 496, 1009, 6562, 3233}
 16 (14- 1)   {18829, 124618, 312962, 578955, 958109, 1340652, 376761, 329340, 537059, 681069, 886898, 1626673, 1665667, 2021413}
 17 ( 2- 2)   {58618, 76438}
 18 ( 6- 1)   {2929, 13154, 4394, 7154, 3283, 4274}
 19 ( 1-13)   {407}
 20 ( 3- 2)   {282595, 824963, 845130}
 21 ( 1-14)   {8208}
 22 ( 2- 3)   {2178, 6514}
 
 23 ( 4- 1)   {10933, 59536, 73318, 50062}
 24 (12- 1)   {5908997, 17347727, 23131558, 17571846, 30442597, 49340036, 44870531, 23070276, 13216291, 44733413, 5981093, 11743403}
 25 ( 1-15)   {1634}
 26 ( 1-16)   {9474}
 
 27 ( 1-17)   {54748}
 28 ( 1-18)   {32164049651}
 29 ( 1-19)   {92727}
 30 ( 1-20)   {93084}
 
 31 (12- 2)   {1680387, 5299971, 15250704, 6611844, 2689794, 12783081, 39326052, 45130596, 45579685, 68505765, 27073124, 11602212}
 32 ( 1-21)   {548834}
 33 ( 1-22)   {4210818}
 34 ( 3- 3)   {2767918, 8807272, 5841646}
 
 35 ( 1-23)   {9926315}
 36 ( 6- 2)   {2191663, 5345158, 2350099, 9646378, 8282107, 5018104}
 37 ( 1-24)   {9800817}
 38 ( 2- 4)   {8139850, 9057586}
 39 ( 1-25)   {1741725}
 40 ( 1-26)   {24678051}
 41 ( 2- 5)   {2755907, 6586433}
 42 ( 1-27)   {472335975}
 43 (10- 2)   {180450907, 564207094, 440329717, 468672187, 369560719, 837322786, 359260756, 451855933, 527799103, 857521513}

 44 ( 1-28)   {24678050}
 45 ( 1-29)   {146511208}
 46 ( 1-30)   {88593477}
 47 ( 7- 1)   {1139785743, 5136409024, 3559173428, 4863700423, 1418899523, 9131926726, 7377037502}
 48 ( 3- 4)   {180975193, 951385123, 525584347}
 49 ( 1-31)   {912985153}
 50 ( 1-32)   {534494836}
 
 51 ( 2- 6)   {277668893, 756738746}
 
 52 ( 1-33)   {32164049650}
 53 ( 7- 2)   {59906808718, 90920874919, 136095696124, 571650873350, 99312318232, 71352591397, 66814785298}
 54 ( 1-34)   {42678290603}
 55 ( 1-35)   {4679307774}
 56 ( 1-36)   {40028394225}
 
 57 ( 1-37)   {49388550606}
 58 ( 1-38)   {82693916578}
 59 ( 1-39)   {94204591914}
 60 ( 5- 1)   {3656948275943, 5759076689801, 6405584099531, 5650346085989, 6213095485028}
 61 ( 1-40)   {44708635679}
 
 62 ( 3- 5)   {3396705890823, 6294418483143, 3654709782417}
 63 ( 9- 1)   {1113928853354, 3643890762383, 3764461348892, 3764592377975, 5486860104254, 1128275756843, 1308860468429, 4217390478269, 5840462013812}

 64 ( 1-41)   {28116440335967}
 65 ( 6- 3)   {22955961974580, 73803590128032, 32357226447319, 24318257549352, 27971919071792, 94220062144011}
 66 ( 4- 2)   {27510477911590, 47800729611562, 28794385423806, 36834169210461}

  -

 67 ( 3- 6)   {255349823145519, 447090882837630, 321411732579837}
 68 ( 4- 3)   {1988938580054728, 5146957705687367, 2276352319249162, 3745072497367240}
 69 ( 1-42)   {4338281769391370}
 70 ( 7- 3)   {2419253396913226, 5564859798630665, 4314122390900936, 5561890395668808, 4840861420987271, 2766744975063429, 3814368015105159}

 71 (13- 1)   {18963633035544997, 52551389500766905, 35876461872431926, 21697619891079652, 69477330558375418, 21897923093961655, 69228536582676925, 38160589126493611, 37909523382771553, 36306344090162179, 33637808638944484, 25950934023321628, 35624633319183334}
 72 ( 2- 7)   {21914086555935085, 37878721692554416}
 73 ( 1-43)   {35641594208964132}
 74 ( 1-44)   {4338281769391371}
 75 ( 1-45)   {21897142587612075}
 
 76 ( 1-46)   {35875699062250035}

 77 ( 2- 8)   {187864919457180831, 375609204308055082}
 78 ( 5- 2)   {1795245058188658603, 1940006654627814913, 2859065497110079396, 5571576991631940457, 4088047402918504546}
 79 ( 5- 3)   {1663709064090943716, 4077790934430101361, 2736510308762619339, 2870463571149040395, 2869264747464021138}
 80 ( 1-47)   {3289582984443187032}
 
 81 ( 1-48)   {1517841543307505039}
 82 ( 1-49)   {4498128791164624869}
 83 ( 1-50)   {4929273885928088826}
 84 ( 2- 9)   {13397885590701080090, 40091536165423401387}

 85 ( 6- 4)   {110626217298381828545, 146915855127127161329, 229223750849314583096, 347285156784769167782, 139948607895288216203, 365753356563451932671}
 86 ( 1-51)   {128468643043731391252}
 87 ( 1-52)   {63105425988599693916}
 88 ( 4- 4)   {129071530839696064382, 347328549450841014962, 237866180836931794379, 357668536389162411710}
 89 ( 5- 4)   {119204730567652576196, 220602798577821749852, 248765172852935962841, 247670499321072742964, 330535042354805755595}

 90 ( 1-53)   {449177399146038697307}
 
 91 (14- 2)   {10636207155096237927550, 17810399355898537585315, 29004806621022790911382, 27798354257924032012720, 18425671021807766709823, 10745681722149970125328, 18989387814955285091968, 47884659771227634888553, 11925507595333861127731, 18399115705999414000444, 44932379349757066271101, 26699880941081568083608, 30133760133393802432549, 18344342589646448004016}
 92 (18- 1)   {10074069541108119620821, 18935428129475061827932, 28415252997720554102092, 27233895488449729771663, 28470768372157693427350, 10181976920394277400584, 27852314084047558510219, 10688610940325001073897, 19553092616993382187366, 36662895609794210663524, 27211241280778321507795, 10180397389391156436853, 28388673910517178592951, 29032929335610790615729, 44371031395529147765128, 18399103996082548276483, 37841895989779904859691, 64484670033935168320603}
 93 ( 6- 5)   {18962820859504701211892, 28978179990182148302144, 37867673425925389700897, 28498148899689232677881, 39640140247078869683999, 46142684925982472355170}

 94 ( 1-54)   {35452590104031691935943}
 95 ( 1-55)   {27879694893054074471405}
 96 ( 1-56)   {27907865009977052567814}
 97 ( 1-57)   {28361281321319229463398}
 98 ( 1-58)   {21887696841122916288858}
 99 ( 4- 5)   {169557221530718698042702, 169562018671994452158988, 338161213991248464720940, 248945121317161275449387}

100 ( 1-59)   {174088005938065293023722}
101 ( 2-10)   {108493282045082839040458, 183144838903847612823687}
102 ( 1-60)   {239313664430041569350093}
103 ( 1-61)   {188451485447897896036875}
104 ( 1-62)   {1553242162893771850669378}
105 ( 1-63)   {1550475334214501539088894}

106 ( 2-11)   {1553298727699254868304830, 2307549584666787613389634}
107 ( 1-64)   {4422095118095899619457938}
108 ( 1-65)   {3706907995955475988644380}
109 ( 2-12)   {1588396116116684818933208, 1662583586865233549781926}
110 ( 1-66)   {3706907995955475988644381}

-

111 ( 1-67)   {121270696006801314328439376}
112 ( 2-13)   {242402817739393059296681797, 295849292210483378481950761}
113 ( 1-68)   {121204998563613372405438066}
114 ( 1-69)   {174650464499531377631639254}
115 ( 1-70)   {177265453171792792366489765}
116 ( 1-71)   {128851796696487777842012787}
117 ( 9- 2)   {1086767104665073071794069358, 1087711604326152142799037806, 1087239298652275489393232318, 2171687973158060309621788895, 1668615408129926023267489583, 1647904917435001854832981579, 2152805184049778117800849152, 1144789403208407365062959581, 1629003705428087305710293501}

-

118 ( 1-72)   {19008174136254279995012734741}
119 ( 1-73)   {19008174136254279995012734740}
120 ( 1-74)   {14607640612971980372614873089}
121 ( 2-14)   {10049035073962232777012913675, 14146559684641908727185026871}
122 ( 1-75)   {23866716435523975980390369295}

-

123 ( 1-76)   {1927890457142960697580636236639}
124 ( 1-77)   {1145037275765491025924292050346}
125 ( 1-78)   {2309092682616190307509695338915}

126 ( 3- 7)   {13893231469464009318515397541007, 13895416448792275685473829744915, 14056081652489244496435933616913}
127 ( 1-79)   {17333509997782249308725103962772}
128 ( 2-15)   {10622389873461838795139919502096, 20921248349728532825757809753362}

129 ( 1-80)   {186709961001538790100634132976990}
130 ( 1-81)   {186709961001538790100634132976991}

131 ( 2-16)   {1411141399999358029488960346019160, 2240239782870943967874495925085635}
132 ( 1-82)   {1122763285329372541592822900204593}

133 ( 1-83)   {12639369517103790328947807201478392}
134 ( 1-84)   {12679937780272278566303885594196922}

-

37-DIGIT UPDATE: A power interruption ended the search. I will not soon resume it.