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A309631
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a(n) is the smallest positive integer divisible by n such that it is possible to strike out a digit from its decimal expansion (apart from trailing zeros) so that the resulting number is nonzero and divisible by n.
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3
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11, 12, 33, 24, 15, 36, 77, 48, 99, 110, 121, 132, 143, 154, 105, 176, 187, 108, 2109, 120, 231, 242, 253, 264, 125, 286, 297, 728, 3219, 330, 341, 352, 363, 374, 315, 396, 4107, 2128, 4329, 240, 451, 462, 473, 484, 405, 2530, 5217, 1344, 5439, 150, 561, 572, 583
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OFFSET
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1,1
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COMMENTS
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The idea of this sequence comes from a problem in the annual Moscow Mathematical Olympiad (MMO) in 2004: problem 3, Level D. The problem asks for a proof that for any positive n, there exists a number m divisible by n such that it is possible to strike out a certain digit d (not a trailing zero) from its decimal expansion so that the number thus obtained will also be divisible by n and nonzero. Here, the sequence proposes to find the smallest such integer m called a(n).
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LINKS
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Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 2000-2005, Problem 3, Level D, 2004, MSRI, 2011, p. 21 and 130/13 (only cover).
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EXAMPLE
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a(6) = 36 because 36 and 6 are divisible by 6, and there is no integer < 36 with this property.
a(19) = 2109 because 2109 = 19*111 and, if we strike out "1", 209 = 19*11 also is divisible by 19, and there is no integer < 2109 with this property.
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MATHEMATICA
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del[n_] := Block[{m = 10^IntegerExponent[n, 10], d}, d = IntegerDigits[n/m]; Table[ FromDigits[ Delete[d, k]] m, {k, Length@d}]]; a[n_] := Block[{k=n, v}, While[! AnyTrue[del[k], # > 0 && Mod[#, n] == 0 &], k += n]; k]; Array[a, 55] (* Giovanni Resta, Sep 22 2019 *)
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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