Suppose there's a highest Champagnat number; call it c.
Start a set of consecutive integers starting at c+1. The total of all digits so far will then be sod(c+1), which is quite smaller than c+1. Keep adding consecutive integers to the sequence, and their sod's to the total of digits. With each consecutive integer that you add, you'll be adding at least one and usually more than one to the total of digits. After n new numbers have been appended to the sequence, the highest number in the sequence will be c+1+n, but the total of digits will have risen by more than n, so the average slope of the line from original total to current total will be larger than the unit slope of the identity of the highest number in the sequence itself, so at some point the total will exceed or match the lowest number in the sequence at that point, yet be lower than the highest number as the newly added sod can't exceed 9 times the length of the number.
Let's say someone claims that 975 is the highest Champagnat number (it is indeed Champagnat as the series from 973 to 1050 has total digits 975. Start at 976; the sod count then starts at 22. By the time you get to adding in sod(1056) you're only up to 972, but by adding in sod(1057) you bring the sod total up to 985, within the range 976 to 1057, and counting that as another Champagnat number.
This does not give the actual next Champagnat number, just a Champagnat number that's higher than any supposed last Champagnat number, as evidenced by the below table of Champagnat numbers through 1000 where 976 happens to be the highest number below 1000 that is not Champagnat, so before 985 come all the numbers 977 through 984 as Champagnat numbers.
champ series
no. first last
1 0 1 (degenerate case as series consists of solely 1 or includes zero)
10 10 13
12 12 14
14 11 14
15 13 15
17 17 18
19 19 22
20 20 24
22 22 25
25 21 25
26 23 26
27 26 28
28 28 31
29 29 33
33 30 35
34 34 37
35 32 36
37 37 40
38 38 42
42 39 44
45 43 47
46 46 49
47 47 51
51 51 56
55 54 58
56 56 60
57 52 57
60 57 62
63 62 67
65 65 69
69 66 71
70 70 76
72 69 75
74 68 74
75 73 78
77 71 77
78 75 80
79 79 85
81 78 84
83 77 83
84 81 87
87 84 89
88 88 94
90 87 93
91 91 97
92 86 92
94 94 100
95 95 104
96 94 101
98 92 98
99 98 112
100 99 115
101 95 105
102 96 107
103 103 118
104 98 113
105 105 119
106 106 121
108 105 120
109 108 124
110 107 123
111 106 122
112 112 127
113 104 120
114 114 128
115 111 127
117 116 130
118 118 133
119 113 128
120 117 132
121 121 136
122 116 131
123 123 137
124 120 136
125 119 135
126 118 134
127 126 139
128 122 137
129 127 141
130 130 145
132 132 146
134 128 143
135 125 139
136 135 148
137 137 150
138 133 147
140 125 140
141 141 155
142 129 145
143 128 144
144 137 151
145 144 157
146 140 155
147 142 156
148 139 154
150 150 164
151 147 160
152 137 152
153 149 163
154 153 166
155 146 159
156 151 165
157 157 169
159 154 167
161 158 171
162 152 166
163 162 175
164 155 168
165 160 174
166 166 178
168 163 176
169 156 169
170 167 179
171 169 182
172 171 184
173 164 177
175 175 187
177 172 185
178 178 190
179 176 188
180 170 184
181 180 193
182 173 186
183 177 189
184 184 196
185 179 192
186 181 194
187 174 187
188 185 197
189 188 202
190 189 205
191 182 195
192 192 210
194 188 203
195 193 213
196 192 211
197 194 215
198 197 220
199 198 223
200 200 227
201 192 212
202 195 217
203 203 228
205 205 229
206 194 216
207 202 228
208 207 232
209 206 231
210 205 230
211 204 229
212 200 228
213 211 236
214 196 220
216 215 238
219 217 240
220 220 244
221 212 237
222 222 245
223 214 238
224 218 242
225 211 237
226 225 247
227 221 245
228 223 246
229 213 238
230 215 239
231 226 248
232 228 250
233 218 243
234 232 254
235 235 256
236 227 249
237 229 252
238 211 238
239 236 257
240 231 254
241 241 262
242 239 260
243 237 258
244 234 256
245 242 263
246 246 266
247 240 262
248 224 248
249 247 267
251 239 261
252 251 271
253 244 265
254 236 258
255 252 272
257 245 266
258 253 273
259 250 271
260 260 279
261 254 274
262 261 280
263 251 272
264 262 281
265 247 268
266 263 282
267 256 276
268 264 283
269 248 269
270 265 284
271 271 289
272 272 290
273 273 291
274 274 292
275 275 293
276 276 294
277 277 295
278 278 296
279 279 297
280 280 298
281 260 281
282 282 300
284 272 291
285 283 303
286 282 301
287 284 305
288 287 310
289 288 313
290 290 317
291 291 318
292 292 319
293 287 311
294 277 296
295 295 325
296 284 306
297 293 322
298 297 328
299 287 312
300 298 330
302 296 327
303 301 335
304 291 319
305 299 333
306 305 337
307 298 331
308 308 339
309 291 320
310 294 325
311 302 336
312 309 341
313 304 337
314 293 324
315 313 344
316 316 346
318 307 339
319 310 343
320 317 347
321 312 344
322 291 322
323 320 350
324 323 352
325 315 346
326 311 344
327 324 353
328 319 349
329 296 329
330 325 354
331 322 352
332 332 359
333 333 360
334 334 361
335 335 362
336 336 363
337 337 364
338 338 365
339 339 366
340 340 367
341 332 360
342 342 368
343 333 361
345 334 362
346 312 346
347 335 363
348 331 360
349 336 364
350 341 368
351 350 376
352 324 355
353 338 366
354 333 362
355 355 379
357 340 368
358 331 361
359 353 378
360 351 377
361 360 385
362 359 384
363 358 383
364 357 382
365 356 381
366 366 389
367 354 379
368 350 377
369 361 386
370 370 394
372 369 393
373 364 388
374 368 392
376 367 391
377 371 395
378 377 400
379 378 403
380 380 407
381 381 408
382 382 409
383 377 401
384 375 398
385 385 415
387 383 412
388 387 418
389 377 402
390 388 420
391 373 397
392 386 417
393 382 411
394 381 409
395 395 431
396 394 429
397 396 433
398 374 398
399 381 410
400 400 439
402 397 435
403 394 430
404 401 440
405 404 442
406 378 406
408 405 443
409 391 427
410 398 437
411 406 444
412 412 448
413 401 441
414 414 449
415 397 436
416 404 443
417 408 446
418 411 448
419 386 419
420 409 447
421 399 439
422 413 449
423 421 456
424 423 457
425 416 452
426 417 453
427 427 460
428 419 455
429 426 459
430 394 433
431 413 450
432 424 458
433 414 451
434 401 443
435 432 465
436 429 463
437 428 462
438 435 467
439 426 460
440 425 459
441 440 472
442 433 466
443 443 474
444 439 471
445 420 457
447 441 473
448 448 478
449 434 467
450 437 469
451 451 481
452 449 479
453 445 476
455 452 482
456 435 468
457 439 472
458 446 477
459 458 487
460 450 481
461 437 470
462 459 488
463 454 484
464 449 480
465 460 489
466 430 466
467 467 495
468 461 490
469 468 496
470 452 483
471 469 497
472 472 499
473 470 498
474 463 492
475 475 505
476 443 476
477 473 502
478 477 508
479 458 488
480 478 510
482 476 507
483 472 501
484 471 499
485 485 521
486 484 519
487 486 523
489 471 500
490 490 529
492 487 525
493 484 520
494 491 531
495 494 535
498 493 534
499 481 517
500 488 527
501 492 533
502 498 541
503 497 539
504 501 545
505 505 547
506 503 546
507 495 537
508 483 520
509 500 545
510 486 525
511 493 535
512 506 548
513 512 553
514 504 547
515 515 555
516 483 521
517 508 550
519 513 554
520 520 559
522 505 548
523 495 538
524 521 560
525 525 563
526 510 553
527 512 554
528 526 564
529 507 550
530 518 558
531 527 565
532 531 568
533 533 569
534 534 570
535 535 571
536 536 572
537 537 573
538 538 574
539 539 575
540 540 576
541 522 562
542 512 555
543 532 569
544 526 565
545 545 579
546 523 563
547 534 571
548 527 566
549 543 578
550 541 577
551 536 573
552 531 569
553 537 574
554 509 554
555 538 575
556 552 586
557 539 576
558 555 588
559 559 592
560 542 578
561 558 591
563 557 590
564 564 596
565 556 589
566 527 567
567 551 586
568 532 571
570 568 600
571 562 595
572 554 588
573 546 582
574 547 583
575 575 611
576 574 609
577 576 613
578 563 596
579 556 590
580 580 619
582 577 615
583 574 610
584 581 621
585 584 625
587 551 587
588 583 624
589 571 607
590 578 617
591 582 623
592 588 631
593 587 629
594 589 633
595 595 640
596 569 605
597 585 627
598 573 610
599 596 642
600 598 645
601 592 637
602 593 638
603 594 639
604 603 649
605 581 623
606 604 650
607 589 634
608 605 651
609 579 620
610 606 652
611 611 656
612 607 653
613 594 640
614 608 654
615 603 650
616 609 655
617 617 660
618 616 659
619 601 649
620 593 639
621 614 658
622 621 664
623 602 650
624 606 653
625 600 649
626 596 644
627 627 668
628 588 634
629 611 657
630 629 670
631 613 658
633 631 672
634 625 667
635 605 653
636 610 657
637 597 646
638 635 675
639 632 673
640 640 679
641 626 668
642 630 672
643 636 676
644 641 680
645 633 674
646 628 670
647 620 665
648 647 685
649 631 673
650 605 654
651 648 686
652 643 682
653 638 678
654 649 687
655 615 661
656 644 683
657 652 689
658 639 679
659 653 690
660 645 684
661 661 697
662 626 669
663 655 692
664 646 685
665 665 701
666 664 699
667 666 703
668 647 686
669 658 695
670 670 709
671 662 698
672 667 705
673 664 700
674 671 711
675 674 715
676 649 688
677 635 677
678 673 714
679 645 685
680 668 707
681 672 713
682 678 721
683 677 719
684 679 723
685 685 730
687 675 717
688 663 700
689 686 732
690 688 735
691 682 727
692 683 728
693 684 729
695 695 744
696 690 738
697 679 724
699 699 749
700 700 751
701 689 737
702 698 748
703 693 742
704 680 726
705 703 753
706 688 736
707 701 752
708 696 746
710 707 756
711 704 754
713 677 722
714 714 761
715 708 757
716 686 734
717 715 762
718 678 724
719 689 738
720 716 763
721 712 760
722 701 753
723 723 768
724 699 751
725 713 761
726 718 765
727 687 736
728 704 755
729 719 766
730 711 760
731 686 735
732 720 767
733 724 769
734 734 777
735 726 771
736 727 772
737 728 773
738 732 776
739 730 775
740 713 762
741 717 765
742 742 784
744 721 768
745 723 769
746 740 783
747 746 787
748 738 781
749 743 785
750 736 779
751 726 772
752 710 761
753 750 791
754 741 784
755 728 774
756 754 794
757 729 775
759 751 792
760 760 799
761 755 795
762 742 785
763 745 787
764 761 801
765 764 805
766 756 796
768 763 804
769 733 778
770 746 788
771 762 803
772 768 811
773 767 809
774 769 813
775 775 820
776 758 798
777 765 807
778 751 793
779 776 822
780 778 825
781 772 817
782 773 818
783 774 819
784 739 784
785 785 834
786 780 828
787 769 814
789 789 839
790 790 841
791 779 827
792 788 838
793 783 832
794 770 816
795 787 837
796 778 826
797 797 849
798 796 848
799 795 847
800 800 854
801 798 851
802 792 844
803 791 843
804 790 842
805 780 829
806 776 824
807 805 857
808 808 859
809 779 828
810 803 856
811 811 862
812 788 839
813 781 831
814 789 841
815 806 858
816 798 852
817 777 826
818 812 863
819 815 865
820 819 868
821 809 861
822 808 860
823 823 871
824 797 851
825 820 869
826 781 832
827 824 872
828 811 863
829 801 856
830 821 870
831 825 873
832 814 865
834 787 839
835 826 874
836 812 864
837 818 868
838 793 847
839 833 879
840 831 878
841 841 886
843 835 881
844 813 865
845 836 882
846 844 888
847 837 883
849 838 884
850 825 874
851 842 887
852 832 879
853 840 886
854 809 863
855 850 894
856 856 898
857 800 857
858 853 896
860 827 876
861 843 888
862 858 901
863 857 899
864 859 903
865 865 910
866 854 897
867 837 884
868 832 880
869 866 912
870 868 915
871 862 907
872 863 908
873 864 909
874 855 898
875 875 924
876 870 918
877 859 904
879 879 929
880 880 931
881 869 917
882 878 928
883 873 922
884 860 906
885 877 927
886 868 916
887 887 939
888 886 938
889 885 937
890 890 944
891 891 945
892 892 946
893 893 947
894 894 948
895 895 949
896 866 914
897 873 923
898 889 943
899 869 918
900 899 955
901 865 913
902 896 951
903 871 921
904 879 931
905 905 959
906 906 960
907 907 961
908 908 962
909 909 963
910 910 964
911 866 915
912 897 953
914 893 948
915 900 957
916 894 949
918 913 966
920 911 965
921 916 968
922 922 973
924 907 962
925 898 955
926 920 972
927 926 976
928 918 970
929 923 974
930 912 966
933 927 977
934 921 973
935 932 981
936 924 975
937 937 985
939 928 978
940 933 982
941 938 986
942 930 980
943 925 976
945 939 987
947 890 947
948 942 989
949 940 988
950 935 984
951 943 990
952 912 967
953 953 998
954 944 991
955 955 1003
956 902 960
957 945 992
958 951 997
960 960 1019
961 957 1009
962 962 1025
963 961 1023
964 963 1027
965 965 1032
966 957 1011
967 960 1021
969 967 1037
970 970 1045
971 962 1026
972 968 1039
973 972 1048
974 944 992
975 973 1050
977 977 1058
978 964 1031
979 979 1063
980 974 1053
981 975 1055
982 973 1051
983 968 1041
984 982 1068
985 976 1057
986 959 1020
987 987 1077
988 984 1072
989 983 1070
990 989 1081
991 964 1033
992 992 1086
993 978 1062
994 969 1045
995 986 1076
996 991 1085
997 961 1027
998 983 1071
999 998 1096
1000 1000 1099
Consider the set of numbers under 1000 that are NOT Champagnat numbers:
2 3 4 5 6 7 8 9 11 13 16 18 21 23 24 30 31 32 36 39 40 41 43 44 48 49 50 52 53 54 58 59 61 62 64 66 67 68 71 73 76 80 82 85 86 89 93 97 107 116 131 133 139 149 158 160 167 174 176 193 204 215 217 218 250 256 283 301 317 344 356 371 375 386 401 407 446 454 481 488 491 496 497 518 521 562 569 581 586 632 686 694 698 709 712 743 758 767 788 833 842 848 859 878 913 917 919 923 931 932 938 944 946 959 968 976
Is the set of non-Champagnat positive integers finite or infinite? They seem to be dwindling in frequency the higher one goes.
DefDbl A-Z
Dim crlf$, totupto(1000000), champ(1000000), champlow(1000), champhigh(1000)
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For i = 1 To 100000
totupto(i) = totupto(i - 1) + sod(i)
high = i: tot = sod(i)
low = high
didOne = 0
For low = high - 1 To 1 Step -1
tot = tot + sod(low)
If tot >= low And tot <= high Then Exit For
DoEvents
Next
Do While tot >= low And tot <= high
' Text1.Text = Text1.Text & Str(tot): didOne = 1
champ(tot) = champ(tot) + 1
If tot <= 1000 Then
champlow(tot) = low: champhigh(tot) = high
End If
low = low - 1: tot = tot + sod(low): If low < 1 Then Exit Do
DoEvents
Loop
If didOne Then Text1.Text = Text1.Text & crlf
DoEvents
If low > highlow Then highlow = low: Text1.Text = Text1.Text & low & crlf
If low > 1000 Then Exit For
Next
Text1.Text = Text1.Text & crlf & crlf
For i = 1 To 1000
DoEvents
If champ(i) Then Text1.Text = Text1.Text & mform(i, "###0") & mform(champlow(i), "#####0") & mform(champhigh(i), "#####0") & crlf
Next
Text1.Text = Text1.Text & crlf & crlf
For i = 1 To 1000
DoEvents
If champ(i) = 0 Then Text1.Text = Text1.Text & Str(i)
Next
Text1.Text = Text1.Text & crlf & crlf
Text1.Text = Text1.Text & crlf & " done"
End Sub
Function mform$(x, t$)
a$ = Format$(x, t$)
If Len(a$) < Len(t$) Then a$ = Space$(Len(t$) - Len(a$)) & a$
mform$ = a$
End Function
Function sod(n)
s$ = LTrim(Str(n))
tot = 0
For i = 1 To Len(s$)
tot = tot + Val(Mid(s$, i, 1))
Next
sod = tot
End Function
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Posted by Charlie
on 2018-01-23 15:00:21 |
a proof
