[seqfan] Re: Is the set of numbers representable by sum of 5 positive cubes in exactly 3 ways finite?

[hv at crypt.org]

"Sean A. Irvine" <sairvin at gmail.com> wrote:
:Hi Hugo,
:
:Thank you for this.  It would seem to suggest that a great many of these
:sequences could be finite (at least for cubes).
:
:Are you able from your results to get a count of the total for each case --
:i.e., simply assuming they become finite and have no further terms after
:the last non-zero value you observe for that number of representations.
:I'm interested in the shape of the curve of number of representations (0,
:1, 2, 3, ...) versus count of values achieving that.  Maybe that would
:match up with some of theoretical considerations.

Sure thing, table below gives (n, count, first, last) for numbers that have
n representations as a sum of 5 positive cubes. Do you want the same for
sums of 2, 3 or 4 cubes, or more?

Hugo
---
0 5439 1 1290740
1 11062 5 1685758
2 15416 157 2243453
3 18984 766 2715466
4 22391 1252 2875694
5 24881 1765 3248572
6 27866 2430 4021474
7 30132 4472 4113590
8 32746 4392 5417708
9 34805 6112 4810342
10 36986 5860 5386658
11 38996 8371 5580068
12 40572 9801 6293606
13 42768 11512 6821726
14 44572 12419 6714634
15 46076 13833 8521762
16 47899 14435 8915764
17 49888 19251 9060980
18 50915 19592 9768694
19 52262 18936 11062012
20 53460 21960 11381422
21 55211 27936 12641386
22 56745 31347 16838042
23 58409 28755 14095471
24 59427 31760 12942914
25 60972 31032 14204398
26 62190 35505 14413100
27 63215 36954 16454902
28 64629 39411 14712386
29 65991 45891 16376945
30 67321 43317 19904638
31 68787 44253 19242652
32 69789 51507 17793932
33 71072 46873 19312150
34 72104 48329 20945444
35 73844 61200 22178858
36 74331 51696 23179910
37 75425 59067 22810729
38 76288 66249 23007686
39 78093 55602 22562410
40 78362 62434 27484340
41 79557 60488 26037310
42 80790 78408 28800356
43 81905 81271 27414706
44 82552 98155 30757298
45 83805 90208 30345620
46 84687 97560 27738230
47 85625 77904 29835572
48 86784 104896 29107598
49 87509 91512 31051436
50 88482 105048 33988630
51 89272 93339 35401756
52 91015 107136 32522828
53 91034 116397 34371356
54 92451 113112 34579948
55 92718 117928 37199578
56 93493 128367 39959582
57 94634 142416 40290484
58 95646 134226 40189630
59 96340 131832 41037412
60 97060 116208 42312478
61 97857 130320 41578420
62 99551 136800 43276532
63 100393 156736 43622618
64 100414 138384 41149084
65 101442 176112 43361708
66 102683 180279 47757047
67 103723 151640 46311746
68 103726 161568 47340878
69 105390 163080 47767226
70 106321 194130 48226334
71 106597 188271 48705332
72 107208 196344 50962964
73 107956 182736 50425700
74 108657 176688 54918175
75 109469 171072 56194996
76 110481 207936 52351988
77 110637 202392 54048956
78 112336 229230 56032574
79 112799 223488 56562844
80 113109 217008 64152517
81 113934 248760 56604866
82 115578 236816 58750484
83 115785 254232 63822434
84 116502 256824 60740186
85 117420 248256 63243850
86 118156 247248 62593330
87 118716 201384 63296636
88 119670 230040 68458351
89 119824 254528 65251498
90 121362 301176 71552884
91 121689 286560 70450718
92 122004 304695 68898596
93 123276 308736 71479174
94 123573 294120 74021090
95 124119 332632 70521412
96 124521 300672 71322836
97 125504 295632 74627150
98 127005 262872 78122902
99 126849 301995 82016428
100 127857 321966 78011726
101 128286 320328 74826518
102 128250 316224 78965770
103 128935 371008 87602918
104 129949 328320 85331012
105 130814 326808 93584426
106 131379 399680 85612306
107 132995 361304 88781108
108 133152 401976 92376050
109 133575 361152 90665068
110 134666 379800 94647982
111 135139 375192 84749054
112 135249 373491 89143366
113 136566 359640 94535402
114 136945 407672 92324804
115 138197 375705 90817610
116 137745 387360 99421556
117 139235 373680 99536684
118 139031 392328 96600452
119 140465 412560 99875282
120 140153 354024 107001778
121 140853 379296 98865338
122 141692 418608 105263132
123 142996 479016 102276850
124 142999 451296 105601730
125 144289 453006 108808492
126 144060 484128 106526030
127 144150 437256 106961972
128 145559 510264 107577418
129 145602 478360 108934150
130 146638 486576 108896476
131 147691 471960 115078730
132 147264 483120 112544302
133 148141 523944 117247910
134 149460 484056 116804740
135 149249 563480 115862630
136 149212 500193 123298682
137 150559 608544 117894452
138 150946 511272 118899274
139 151944 517320 117491036
140 151572 497232 128356358
141 152610 516888 120760042
142 153640 597024 127070060
143 152729 590472 126823190
144 153848 553041 131908478
145 154974 575856 134503772
146 154496 530928 143908466
147 156226 563688 136506604
148 156183 594567 134364100
149 157887 602064 131082458
150 158689 627544 150847853
151 158195 629208 147926740
152 159033 646920 132615158
153 158774 654480 140535356
154 160546 648432 145353442
155 160099 555192 146711884
156 160561 693064 140539172
157 161827 700272 144961934
158 162250 558657 144943916
159 162467 702792 146476174
160 162996 589464 145737644
161 164168 641304 148769933
162 164811 700344 146303924
163 164293 720000 155484338
164 164902 686232 144726530
165 165184 741096 145091804
166 166875 655992 151611044
167 166530 687933 153544820
168 166981 728064 163306715
169 167908 707643 161136742
170 168046 652744 177439028
171 169377 662472 172642964
172 168841 766592 156473564
173 170564 689688 174732610
174 170902 693792 167849708
175 170868 776385 170015170
176 172168 816696 164407828
177 172701 852480 167128988
178 172446 798552 173808518
179 173605 831888 175639594
180 173710 701784 182241644
181 174142 720936 171354326
182 174412 822312 179201074
183 174928 928936 179946806
184 175192 804168 175643806
185 176437 832896 182457860
186 175973 856584 177972314
187 177717 811728 190623290
188 176930 907200 179321090
189 177569 845424 183698294
190 178175 884736 188562686
191 179503 793584 189234994
192 179855 733104 201840340
193 179864 884304 193784890
194 179745 904896 183935804
195 180480 951768 188571812
196 181157 949248 207653134
197 181915 945216 192988562
198 182878 937224 198687164
199 182876 916488 190908419
200 183694 969912 200959538
201 183646 1014840 191067188
202 184992 890352 210163460
203 185125 964683 207160798
204 185346 1022904 203418878
205 185516 916560 210567838
206 185636 1099944 212109404
207 186412 1009800 231825542
208 186361 955872 216393476
209 187963 1008216 226699798
210 188806 943704 211760770
211 188716 929664 217886162
212 188842 982080 213215756
213 189572 1048104 225607226
214 190400 1047600 226763654
215 190563 1042560 228089732
216 191603 1061640 213706922
217 191242 1039338 243436154
218 191702 1021896 226171634
219 192158 1107072 240956950
220 192639 1081296 246397190
221 192792 1082808 220330058
222 193246 1102464 238363970
223 194162 1165536 231763298
224 194193 981072 237344692
225 195049 1061208 245854570
226 194618 1040040 238362826
227 195618 1244160 241987252
228 196249 1075248 234290290
229 196410 1132704 246422956
230 196737 1214487 236775604
231 198073 1157976 244255414
232 196926 1147392 237581474
233 199080 1207296 245709230
234 199132 1262808 262601950
235 198681 1120608 253062940
236 200602 1237536 250906604
237 199936 1073808 260885300
238 199821 1094400 280631858
239 200857 1218888 256905779
240 201893 1277424 254212348
241 201410 1355472 251060108
242 202310 1304064 257796292
243 202880 1259712 260714506
244 203112 1216944 260455442
245 203343 1284408 260760812
246 203628 1354536 264029090
247 204602 1357056 278323474
248 205237 1238112 263612102
249 205198 1329336 259027834
250 205998 1428048 268835242
251 205753 1420992 281681348
252 206874 1499688 275719756
253 206737 1434168 279466772
254 207345 1238544 284290682
>254 (inf) 1291464 (inf)