

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.


2



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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).


LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Roman Fedorov, Alexei Belov, Alexander Kovaldzhi, Ivan Yashchenko, Moscow Mathematical Olympiads, 20002005,Problem 3, Level D, 2004, MSRI, 2011, p. 21 and 130/13 (only cover).


EXAMPLE

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.


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 *)


CROSSREFS

Cf. A061760.
Sequence in context: A136433 A172173 A061760 * A075559 A080138 A244068
Adjacent sequences: A309628 A309629 A309630 * A309632 A309633 A309634


KEYWORD

nonn,base


AUTHOR

Bernard Schott, Sep 22 2019


EXTENSIONS

More terms from Giovanni Resta, Sep 22 2019


STATUS

approved



