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Iterating sum of proper divisors function

If s(n) denotes the sum of the proper (less than ‘n’) divisors of n, then s(6) = 6 because 6 is a perfect number:

divisors of 6 = {1, 2, 3, 6}  and 1+2+3 = 6.

 

If we repeat the “s operation” over and over, sometimes the iterates wander a long time, such as if we start with

n =  276, as I wrote in the newsgroup sci.math today, Friday 27 Dec. 2013.

 

But some numbers wander quite high and then reach 1 , for which we may say s(1) =0  , or alter just a bit our definition

of s() so that s(1):= 1.

 

Anyway, n = 702 takes about 300 iterations of s(.) before it reaches 1:

 

Code in PARI/gp with output:

? ccc=702;for(Y=1,300, if(ccc>1, ccc = new(ccc));print(Y,” “, ccc))
1 978
2 990
3 1818
4 2160
5 5280
6 12864
7 21680
8 28912
9 31848
10 47832
11 71808
12 148512
13 359520
14 946848
15 1895712
16 4539360
17 12180336
18 23781648
19 44267568
20 76111632
21 139130668
22 104348008
23 92030552
24 80526748
25 62286692
26 55099864
27 51042536
28 47249404
29 35492780
30 39042100
31 45679474
32 22839740
33 36962884
34 38008124
35 53126668
36 54930932
37 54930988
38 60686612
39 62153644
40 62153700
41 155582364
42 259304164
43 266871836
44 327310564
45 327768476
46 327768532
47 396478124
48 397718356
49 398447980
50 557827508
51 557827564
52 583593332
53 645414448
54 787922384
55 739137616
56 953968304
57 894345316
58 774937460
59 1009670092
60 763542524
61 617043844
62 462782890
63 370226330
64 300597094
65 254298266
66 127318438
67 78349850
68 67380964
69 67381020
70 170849700
71 455276892
72 785095332
73 1353047388
74 2730005796
75 4680011532
76 8989919988
77 15034639116
78 25057732084
79 30907073036
80 33007287796
81 34078491020
82 50530186420
83 70844827340
84 99182758612
85 106237222700
86 165551444212
87 195690975116
88 228113158756
89 254901208604
90 263632807396
91 274851227420
92 507715459300
93 751418881500
94 1749303191076
95 3480682068444
96 6604825822756
97 7467599613404
98 7483271439364
99 7483271439420
100 19285028182980
101 48485555197884
102 91621492377156
103 191982873147324
104 371492151175236
105 719488350665532
106 1652151220493508
107 3368985514491324
108 6306049809179204
109 6306049809179260
110 8828469732851300
111 13066135204621660
112 18405797678228324
113 21945449165668756
114 26323036089475244
115 27487354482052276
116 27492212193118540
117 38489217056547572
118 39864013609492492
119 41491116205801108
120 49034955515947436
121 58804502365944916
122 71372078274996884
123 93526580908250476
124 101659327074186644
125 111202674691272556
126 112182433939666580
127 158408361492762220
128 225535865174877908
129 232980109946018092
130 241300936709076188
131 260041794337473316
132 282654124279863644
133 341008402262784676
134 364526223108495644
135 368599141802455396
136 368822249063390108
137 370877736802670692
138 377810030948521052
139 398232194783578372
140 398239194643966844
141 408534386130917956
142 486602641614267836
143 486602641614267892
144 486602641614267948
145 934231026364583892
146 1557425833260726828
147 2595709978031227092
148 4326183296718712044
149 9780311870838919956
150 18808292059305637644
151 36664297198873376004
152 71167230181785000564
153 147202422388016680076
154 148262320923436016500
155 225062240123098506380
156 346174744671040944820
157 499500630340013927156
158 517339938566442996382
159 382819168719052189538
160 198967549610148982942
161 100687628661143619938
162 52977517522207772062
163 28835699200846055906
164 14637410761850790394
165 9367769454714031622
166 4689767177457166378
167 3585473909277449822
168 1807765549154014402
169 905718271755613598
170 455774796192205522
171 280241577632419118
172 140127125867744482
173 70777748619665630
174 66186575354217250
175 58026787585796942
176 36926437581869410
177 30375754312242590
178 24300603449794090
179 19503269881071830
180 20617747263830890
181 17500144273783910
182 15038477361839962
183 7742598563978990
184 6351144328711090
185 5081454177821990
186 4112555142883258
187 2157405976594682
188 1374892011310726
189 687446005655366
190 388946435248954
191 248809136911046
192 126730645933114
193 67159439664326
194 41148951138874
195 20574948398426
196 10514229373774
197 7525127376242
198 5393061148558
199 2696530574282
200 1586194455514
201 793097227760
202 1069863649696
203 1046129909984
204 1013438350360
205 1266797938040
206 1584372307960
207 2481391821320
208 3102378418000
209 4531341206960
210 6048108182800
211 8491829678180
212 9609470104540
213 10586195032100
214 12594247949596
215 9445685962204
216 8483499879524
217 6367542494476
218 5531758280564
219 4291487534476
220 3242783063924
221 2432541702124
222 1828322519700
223 3461623971500
224 4584759647956
225 3441520593324
226 6049222898724
227 9766324922076
228 14920774186596
229 21311255202204
230 32936265266532
231 45162282364572
232 63620610281828
233 47715457711378
234 26969606532590
235 22240689224098
236 12876188498222
237 9197277498754
238 4599035071886
239 2299610540098
240 1231080512318
241 698681894338
242 350182749662
243 175091374834
244 97917614606
245 70970965618
246 50790212942
247 37843701322
248 29875397558
249 15169891570
250 12135913274
251 6067956640
252 8267591300
253 11099950348
254 8346106212
255 12174757788
256 16233010412
257 12252165748
258 9189124318
259 6741790226
260 3436558318
261 1718279162
262 874165798
263 562709978
264 281354992
265 263770336
266 269564984
267 274749136
268 257577346
269 164722238
270 82623250
271 73052270
272 61500610
273 49200506
274 25008538
275 12504272
276 15083248
277 15149192
278 13538308
279 14022218
280 10015894
281 7460990
282 5968810
283 4804190
284 3843370
285 3337154
286 1790794
287 952694
288 476350
289 536978
290 376942
291 222818
292 111412
293 122444
294 122500
295 189119
296 27025
297 8687
298 1969
299 191
300 1

 

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Written by meditationatae

December 27, 2013 at 5:48 pm

Posted in History

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