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.

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

Offset: 1

Author

Bernard Schott, Sep 22 2019

Keywords

• nonn
• base

Comments

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).

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• 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).

Crossrefs

Cf. A061760.

Programs

Mathematica

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 *)

Extensions

More terms from Giovanni Resta, Sep 22 2019