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A309631 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.

%I A309631 #51 Oct 01 2020 03:33:16
%S A309631 11,12,33,24,15,36,77,48,99,110,121,132,143,154,105,176,187,108,2109,
%T A309631 120,231,242,253,264,125,286,297,728,3219,330,341,352,363,374,315,396,
%U A309631 4107,2128,4329,240,451,462,473,484,405,2530,5217,1344,5439,150,561,572,583
%N A309631 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.
%C A309631 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).
%H A309631 Rémy Sigrist, <a href="/A309631/b309631.txt">Table of n, a(n) for n = 1..10000</a>
%H A309631 Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, <a href="https://bookstore.ams.org/mcl-7/">Moscow Mathematical Olympiads, 2000-2005</a>, Problem 3, Level D, 2004, MSRI, 2011, p. 21 and 130/13 (only cover).
%e A309631 a(6) = 36 because 36 and 6 are divisible by 6, and there is no integer < 36 with this property.
%e A309631 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.
%t A309631 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 *)
%Y A309631 Cf. A061760.
%K A309631 nonn,base
%O A309631 1,1
%A A309631 _Bernard Schott_, Sep 22 2019
%E A309631 More terms from _Giovanni Resta_, Sep 22 2019