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Quite a triplet (Posted on 2015-07-02) Difficulty: 3 of 5
Let (n,n+1,n+2) represent a triplet of consecutive numbers, each having 5 distinct prime factors.

Find tne value of the lowest n.

No Solution Yet Submitted by Ady TZIDON    
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re(2): computer solution -- prolonged search | Comment 8 of 10 |
(In reply to re: computer solution by broll)

broll asks if when the search is extended, whether we'll get to triplets where each number has the five distinct factors unrepeated (not squared, cubed, etc.). As noted previously, that would necessarily entail the first and last being odd, so as not to include a factor of 4.  The following list shows solutions which have the sequence begin and end on odd numbers, and in some of these instances, such as the one starting at 16467033, none of the primes is raised to a power higher than 1:

13680665 13680666 13680667
5 5 5
 5 * 17 * 19 * 43 * 197      2 * 3^2 * 47 * 103 * 157      7 * 11 * 13 * 79 * 173     

16467033 16467034 16467035
5 5 5
 3 * 11 * 17 * 149 * 197      2 * 19 * 23 * 83 * 227      5 * 13 * 37 * 41 * 167     

16598645 16598646 16598647
5 5 5
 5 * 7 * 41 * 43 * 269      2 * 3^2 * 53 * 127 * 137      13 * 17 * 19 * 59 * 67     

17065839 17065840 17065841
5 5 5
 3 * 7 * 23 * 89 * 397      2^4 * 5 * 11^2 * 41 * 43      13 * 17 * 31 * 47 * 53     

17168723 17168724 17168725
5 5 5
 11 * 13 * 19 * 71 * 89      2^2 * 3^2 * 47 * 73 * 139      5^2 * 7 * 17 * 29 * 199     

17380363 17380364 17380365
5 5 5
 7 * 11 * 13 * 97 * 179      2^2 * 19 * 23 * 61 * 163      3 * 5 * 47 * 89 * 277     

18185869 18185870 18185871
5 5 5
 13 * 17 * 19 * 61 * 71      2 * 5 * 23 * 37 * 2137      3 * 11 * 29 * 31 * 613     

18371065 18371066 18371067
5 5 5
 5 * 29 * 31 * 61 * 67      2 * 7 * 23 * 59 * 967      3 * 11^2 * 13 * 17 * 229     

18600295 18600296 18600297
5 5 5
 5 * 7 * 17 * 43 * 727      2^3 * 11 * 13 * 71 * 229      3 * 19 * 47 * 53 * 131     

18776483 18776484 18776485
5 5 5
 11 * 17 * 31 * 41 * 79      2^2 * 3^2 * 19 * 97 * 283      5 * 7 * 13 * 29 * 1423     

19933639 19933640 19933641
5 5 5
 11 * 17 * 37 * 43 * 67      2^3 * 5 * 23 * 47 * 461      3^3 * 7^2 * 13 * 19 * 61     

21134553 21134554 21134555
5 5 5
 3 * 11 * 17 * 101 * 373      2 * 7 * 79 * 97 * 197      5 * 13 * 19 * 109 * 157     

21374353 21374354 21374355
5 5 5
 7 * 11 * 13 * 131 * 163      2 * 19 * 43 * 103 * 127      3 * 5 * 17 * 109 * 769     

21623523 21623524 21623525
5 5 5
 3 * 31 * 41 * 53 * 107      2^2 * 13 * 17 * 61 * 401      5^2 * 7 * 11 * 47 * 239     

21871365 21871366 21871367
5 5 5
 3 * 5 * 29 * 137 * 367      2 * 11 * 37 * 97 * 277      7 * 17 * 23 * 61 * 131     

22247553 22247554 22247555
5 5 5
 3 * 29 * 31 * 73 * 113      2 * 7 * 61 * 109 * 239      5 * 11 * 23 * 43 * 409     

22412533 22412534 22412535
5 5 5
 11 * 13 * 19 * 73 * 113      2 * 23 * 29 * 53 * 317      3 * 5 * 31 * 157 * 307     

22721585 22721586 22721587
5 5 5
 5 * 23 * 41 * 61 * 79      2 * 3 * 47 * 197 * 409      7 * 19 * 29 * 43 * 137     

24845313 24845314 24845315
5 5 5
 3 * 17 * 23 * 59 * 359      2 * 13 * 43 * 71 * 313      5 * 11 * 29 * 37 * 421     

24862563 24862564 24862565
5 5 5
 3^2 * 11 * 23 * 61 * 179      2^2 * 19 * 41 * 79 * 101      5 * 7 * 13 * 53 * 1031     

25118093 25118094 25118095
5 5 5
 7 * 11 * 13 * 23 * 1091      2 * 3 * 101 * 181 * 229      5 * 17 * 19 * 103 * 151     

25228929 25228930 25228931
5 5 5
 3 * 11 * 61 * 83 * 151      2 * 5 * 23 * 229 * 479      7 * 13 * 37 * 59 * 127     

25325573 25325574 25325575
5 5 5
 7 * 13 * 53 * 59 * 89      2 * 3 * 31 * 47 * 2897      5^2 * 11 * 19 * 37 * 131     

25345333 25345334 25345335
5 5 5
 13 * 23 * 29 * 37 * 79      2 * 7 * 17 * 109 * 977      3 * 5 * 19 * 113 * 787     

25596933 25596934 25596935
5 5 5
 3 * 19 * 37 * 53 * 229      2 * 11 * 17 * 89 * 769      5 * 7 * 13 * 101 * 557     

26217245 26217246 26217247
5 5 5
 5 * 19 * 41 * 53 * 127      2 * 3 * 11 * 163 * 2437      7 * 17 * 29 * 71 * 107     

26285985 26285986 26285987
5 5 5
 3^3 * 5 * 11 * 31 * 571      2 * 43 * 53 * 73 * 79      7 * 13 * 19 * 23 * 661     

26296359 26296360 26296361
5 5 5
 3 * 29 * 47 * 59 * 109      2^3 * 5 * 23 * 101 * 283      7 * 13 * 19 * 67 * 227     

26520219 26520220 26520221
5 5 5
 3^2 * 11 * 19 * 23 * 613      2^2 * 5 * 47 * 89 * 317      7^2 * 13 * 17 * 31 * 79     

26578615 26578616 26578617
5 5 5
 5 * 7 * 59 * 61 * 211      2^3 * 17 * 23 * 29 * 293      3 * 13 * 37 * 113 * 163     

27140113 27140114 27140115
5 5 5
 7 * 11 * 13 * 19 * 1427      2 * 29 * 41 * 101 * 113      3 * 5 * 23 * 97 * 811 

  Posted by Charlie on 2015-07-03 11:36:27
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