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Ulam numbers (Posted on 2017-06-03) Difficulty: 3 of 5
We define recursively the Ulam numbers by setting u1 = 1, u2 = 2, and for each subsequent integer n, we set n equal to the next Ulam number if it can be written uniquely as the sum of two different Ulam numbers; e.g.: u3 = 3, u4 = 4, u5 = 6, etc.

Prove that there are infinitely many Ulam numbers.

Now a D4 BONUS.
3 (=1+2). Find another Ulam number is that is the sum of two consecutive Ulam numbers.

How about a 3rd one?

See The Solution Submitted by Ady TZIDON    
Rating: 3.0000 (1 votes)

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Solution The proof and the bonus | Comment 2 of 5 |
Proof by contradiction:

Suppose there were a largest Ulam number. The sum of the largest and the second largest would be a number that, thanks to the lack of intervening numbers, and the fact that it is the largest sum that could be formed from just two Ulam numbers (of that presumed finite set), would be another Ulam number. QED. Of course the actual presence of other, intervening, Ulam numbers could prevent the hypothesized number from actually being an Ulam number, but of course those numbers themselves would be the counterargument to the "last" Ulam number being the last Ulam number.

Bonus:

The below are the 827 (except the first two are not shown) Ulam numbers below 10,000. Only two were found that are the sum of two consecutive Ulam numbers: 3 = 2 + 1 and 131 = 69 + 62.

Shown are the ordinal position and the Ulam number itself. In the two cases mentioned, the consecutive Ulam numbers that make it up are shown after an asterisk.

3 3 * 2 1
4 4
5 6
6 8
7 11
8 13
9 16
10 18
11 26
12 28
13 36
14 38
15 47
16 48
17 53
18 57
19 62
20 69
21 72
22 77
23 82
24 87
25 97
26 99
27 102
28 106
29 114
30 126
31 131 * 69 62
32 138
33 145
34 148
35 155
36 175
37 177
38 180
39 182
40 189
41 197
42 206
43 209
44 219
45 221
46 236
47 238
48 241
49 243
50 253
51 258
52 260
53 273
54 282
55 309
56 316
57 319
58 324
59 339
60 341
61 356
62 358
63 363
64 370
65 382
66 390
67 400
68 402
69 409
70 412
71 414
72 429
73 431
74 434
75 441
76 451
77 456
78 483
79 485
80 497
81 502
82 522
83 524
84 544
85 546
86 566
87 568
88 585
89 602
90 605
91 607
92 612
93 624
94 627
95 646
96 668
97 673
98 685
99 688
100 690
101 695
102 720
103 722
104 732
105 734
106 739
107 751
108 781
109 783
110 798
111 800
112 820
113 847
114 849
115 861
116 864
117 866
118 891
119 893
120 905
121 927
122 949
123 983
124 986
125 991
126 1018
127 1020
128 1023
129 1025
130 1030
131 1032
132 1035
133 1037
134 1052
135 1079
136 1081
137 1101
138 1103
139 1125
140 1155
141 1157
142 1164
143 1167
144 1169
145 1186
146 1191
147 1208
148 1230
149 1252
150 1257
151 1296
152 1308
153 1311
154 1313
155 1335
156 1338
157 1340
158 1355
159 1360
160 1377
161 1387
162 1389
163 1404
164 1406
165 1428
166 1431
167 1433
168 1462
169 1465
170 1470
171 1472
172 1489
173 1492
174 1509
175 1514
176 1516
177 1531
178 1536
179 1538
180 1550
181 1553
182 1594
183 1602
184 1604
185 1616
186 1641
187 1643
188 1646
189 1648
190 1660
191 1682
192 1707
193 1709
194 1721
195 1724
196 1748
197 1765
198 1770
199 1790
200 1792
201 1812
202 1814
203 1834
204 1836
205 1853
206 1856
207 1858
208 1900
209 1902
210 1919
211 1941
212 1944
213 1946
214 1966
215 1968
216 1985
217 2010
218 2012
219 2032
220 2034
221 2054
222 2056
223 2090
224 2093
225 2095
226 2112
227 2115
228 2117
229 2134
230 2156
231 2178
232 2247
233 2249
234 2252
235 2254
236 2288
237 2327
238 2330
239 2332
240 2354
241 2371
242 2393
243 2418
244 2420
245 2445
246 2447
247 2462
248 2464
249 2481
250 2484
251 2486
252 2511
253 2513
254 2525
255 2550
256 2552
257 2572
258 2574
259 2581
260 2584
261 2589
262 2613
263 2616
264 2618
265 2628
266 2630
267 2633
268 2635
269 2650
270 2660
271 2662
272 2674
273 2696
274 2721
275 2723
276 2748
277 2750
278 2762
279 2787
280 2789
281 2809
282 2811
283 2814
284 2816
285 2831
286 2833
287 2897
288 2899
289 2916
290 2919
291 2921
292 2985
293 2987
294 3029
295 3031
296 3038
297 3041
298 3043
299 3065
300 3068
301 3070
302 3085
303 3090
304 3092
305 3107
306 3109
307 3131
308 3153
309 3205
310 3207
311 3214
312 3217
313 3219
314 3236
315 3239
316 3261
317 3263
318 3288
319 3290
320 3305
321 3368
322 3371
323 3373
324 3390
325 3393
326 3395
327 3415
328 3417
329 3451
330 3454
331 3456
332 3473
333 3476
334 3481
335 3483
336 3495
337 3525
338 3527
339 3544
340 3547
341 3549
342 3591
343 3593
344 3605
345 3608
346 3610
347 3622
348 3625
349 3630
350 3632
351 3649
352 3669
353 3671
354 3691
355 3696
356 3698
357 3723
358 3725
359 3740
360 3742
361 3759
362 3762
363 3764
364 3806
365 3808
366 3825
367 3872
368 3874
369 3886
370 3916
371 3918
372 3930
373 3952
374 3960
375 3962
376 3974
377 3991
378 3994
379 4018
380 4038
381 4040
382 4057
383 4101
384 4118
385 4121
386 4148
387 4150
388 4153
389 4155
390 4165
391 4167
392 4187
393 4211
394 4233
395 4258
396 4260
397 4294
398 4297
399 4324
400 4326
401 4341
402 4343
403 4363
404 4365
405 4368
406 4370
407 4390
408 4392
409 4404
410 4407
411 4409
412 4451
413 4453
414 4470
415 4517
416 4519
417 4531
418 4534
419 4536
420 4578
421 4580
422 4600
423 4602
424 4619
425 4622
426 4624
427 4641
428 4644
429 4646
430 4666
431 4668
432 4707
433 4729
434 4732
435 4734
436 4754
437 4756
438 4798
439 4800
440 4878
441 4881
442 4883
443 4900
444 4903
445 4905
446 4925
447 4927
448 4969
449 4971
450 4996
451 4998
452 5018
453 5020
454 5032
455 5035
456 5037
457 5049
458 5052
459 5057
460 5059
461 5079
462 5081
463 5096
464 5118
465 5159
466 5162
467 5164
468 5181
469 5184
470 5186
471 5206
472 5208
473 5250
474 5252
475 5269
476 5272
477 5274
478 5291
479 5294
480 5296
481 5316
482 5318
483 5335
484 5357
485 5382
486 5384
487 5484
488 5487
489 5489
490 5514
491 5516
492 5531
493 5533
494 5550
495 5597
496 5599
497 5616
498 5663
499 5665
500 5685
501 5687
502 5765
503 5768
504 5770
505 5795
506 5797
507 5812
508 5814
509 5826
510 5829
511 5853
512 5856
513 5858
514 5873
515 5875
516 5878
517 5880
518 5895
519 5900
520 5902
521 5944
522 5946
523 6024
524 6027
525 6029
526 6046
527 6049
528 6051
529 6068
530 6107
531 6110
532 6112
533 6134
534 6154
535 6156
536 6176
537 6178
538 6239
539 6242
540 6244
541 6283
542 6308
543 6310
544 6322
545 6325
546 6327
547 6352
548 6354
549 6366
550 6391
551 6393
552 6410
553 6418
554 6420
555 6435
556 6437
557 6457
558 6459
559 6479
560 6481
561 6520
562 6523
563 6525
564 6550
565 6552
566 6567
567 6569
568 6586
569 6633
570 6635
571 6652
572 6721
573 6723
574 6735
575 6738
576 6740
577 6782
578 6784
579 6804
580 6806
581 6831
582 6833
583 6862
584 6865
585 6870
586 6872
587 6889
588 6892
589 6894
590 6909
591 6911
592 6914
593 6916
594 6931
595 6936
596 6938
597 6980
598 6982
599 7038
600 7041
601 7043
602 7060
603 7063
604 7065
605 7082
606 7104
607 7156
608 7158
609 7170
610 7192
611 7195
612 7197
613 7217
614 7219
615 7258
616 7275
617 7278
618 7280
619 7297
620 7300
621 7302
622 7322
623 7324
624 7366
625 7368
626 7424
627 7427
628 7432
629 7434
630 7459
631 7461
632 7471
633 7473
634 7476
635 7478
636 7490
637 7498
638 7500
639 7520
640 7522
641 7537
642 7539
643 7559
644 7578
645 7581
646 7583
647 7600
648 7603
649 7605
650 7622
651 7691
652 7693
653 7727
654 7730
655 7735
656 7737
657 7749
658 7752
659 7754
660 7774
661 7776
662 7779
663 7781
664 7798
665 7818
666 7820
667 7881
668 7884
669 7886
670 7928
671 7930
672 7950
673 7952
674 7977
675 7979
676 8008
677 8011
678 8016
679 8018
680 8055
681 8057
682 8060
683 8062
684 8077
685 8096
686 8099
687 8101
688 8126
689 8128
690 8143
691 8145
692 8165
693 8167
694 8192
695 8194
696 8236
697 8238
698 8250
699 8253
700 8255
701 8272
702 8275
703 8277
704 8294
705 8316
706 8338
707 8368
708 8370
709 8404
710 8421
711 8424
712 8426
713 8451
714 8453
715 8468
716 8487
717 8509
718 8512
719 8514
720 8531
721 8539
722 8541
723 8553
724 8570
725 8573
726 8617
727 8619
728 8639
729 8666
730 8668
731 8702
732 8705
733 8749
734 8751
735 8754
736 8756
737 8771
738 8773
739 8812
740 8815
741 8817
742 8842
743 8844
744 8856
745 8873
746 8876
747 8917
748 8942
749 8947
750 8949
751 8964
752 8966
753 8969
754 8971
755 8983
756 9013
757 9015
758 9027
759 9052
760 9054
761 9071
762 9093
763 9118
764 9120
765 9132
766 9135
767 9137
768 9162
769 9164
770 9179
771 9184
772 9186
773 9193
774 9196
775 9240
776 9250
777 9252
778 9262
779 9308
780 9311
781 9313
782 9355
783 9357
784 9377
785 9379
786 9399
787 9401
788 9443
789 9445
790 9509
791 9511
792 9553
793 9555
794 9575
795 9577
796 9619
797 9621
798 9641
799 9643
800 9663
801 9665
802 9699
803 9702
804 9704
805 9721
806 9724
807 9726
808 9743
809 9765
810 9804
811 9807
812 9809
813 9834
814 9836
815 9851
816 9853
817 9870
818 9917
819 9919
820 9939
821 9941
822 9958
823 9961
824 9963
825 9980
826 9983
827 9985

  Posted by Charlie on 2017-06-03 14:31:38
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