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%I #20 May 02 2024 10:49:39
%S 10,10,30,102,120,240,1260,1680,8596,34580,113760,576840,3579840,
%T 14938560,109133640
%N The smallest number k that has a factorization k = f1*f2*...*fr where the numbers k, f1, f2, ..., fr together contain every number from 0 to n, without overlap, as substrings.
%C In the first thirteen terms the 'perfect' solutions (ones without any excess digits) are for n = 6, 9, 10, 11, 12. It is likely such solutions become very rare as n increases.
%e a(1) = 10 as 10 = 2 * 5, and {10, 2, 5} contains the numbers 0 and 1 as non-overlapping substrings, and no smaller number has a similar factorization.
%e a(2) = 10 as {10, 2, 5} also contains the number 0, 1, and 2.
%e a(3) = 30 as 30 = 2 * 15, and {30, 2, 15} contains 0,..,3.
%e a(4) = 102 as 102 = 3 * 34, which contains 0,..,4.
%e a(5) = 120 as 120 = 2 * 3 * 4 * 5, which contains 0,..,5.
%e a(6) = 240 as 240 = 3 * 5 * 16, which contains 0,..,6. The first perfect solution.
%e a(7) = 1260 as 1260 = 3 * 3 * 4 * 5 * 7, which contains 0,..,7.
%e a(8) = 1680 as 1680 = 2 * 2 * 3 * 4 * 5 * 7, which contains 0,..,8.
%e a(9) = 8596 as 8596 = 2 * 14 * 307, which contains 0,..,9. A perfect solution.
%e a(10) = 34580 = 7 * 10 * 19 * 26, which contains 0,..,10. A perfect solution. Note that all three of 0, 1, and 10 must appear as separate nonoverlapping substrings.
%e a(11) = 113760 as 113760 = 2 * 4 * 9 * 10 * 158, which contains 0,..,11. A perfect solution.
%e a(12) = 576840 as 576840 = 10 * 11 * 12 * 19 * 23, which contains 0,..,12. A perfect solution.
%e a(13) = 3579840 as 3579840 = 2 * 2 * 6 * 10 * 11 * 12 * 113, which contains 0,..,13.
%e a(14) = 14938560 as 14938560 = 7 * 10 * 12 * 12 * 13 * 114, which contains 0,...,14. A perfect solution.
%e a(15) = 109133640 as 109133640 = 2 * 11 * 14 * 18 * 127 * 155, which contains 0,...,15.
%Y Cf. A027746, A001055, A370970,
%K nonn,base,more
%O 1,1
%A _Scott R. Shannon_, Apr 14 2024
%E a(14)-a(15) from _David Consiglio, Jr._, Apr 25 2024
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