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a(n) is the least k such that the decimal representation of k*n contains at least two digits which are the same.
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%I #32 Sep 18 2020 02:09:36

%S 11,11,11,11,11,11,11,11,11,10,1,12,9,8,15,7,7,8,6,5,12,1,5,6,4,13,37,

%T 4,4,10,5,7,1,8,13,4,3,3,3,5,13,6,8,1,5,7,3,3,7,2,5,13,4,20,1,2,2,2,2,

%U 5,2,7,4,7,7,1,5,4,6,10,5,2,4,3,3,3,1,11,6,5,14,8,2,3,3

%N a(n) is the least k such that the decimal representation of k*n contains at least two digits which are the same.

%C First differs from A045538 at a(21) = 12 since 21 * 12 = 252 contains two equal but not consecutive digits. A045538(21) = 16.

%C The largest value is a(27)=37. - _Robert Israel_, Sep 17 2020

%H Robert Israel, <a href="/A337241/b337241.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F a(n) <= A045538(n).

%F a(n) = 1 for any n >= 10^10. - _Rémy Sigrist_, Sep 15 2020

%e a(21) = 12 because 21*12 = 252 is the smallest multiple of 21 with equal digits.

%e a(23) = 5 because 23*5 = 115 is the smallest multiple of 23 with equal digits.

%e a(34) = 8 because 34*8 = 272 is the smallest multiple of 34 with equal digits.

%p f:= proc(n) local k,L;

%p for k from 1 do

%p L:= convert(k*n,base,10);

%p if nops(convert(L,set))<nops(L) then return k fi

%p od;

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Sep 17 2020

%t a[n_] := Module[{k = 1}, While[Max@ (Last /@ Tally @ IntegerDigits[k*n]) == 1, k++]; k]; Array[a, 100] (* _Amiram Eldar_, Aug 22 2020 *)

%o (PARI) a(n) = {my(k=1, d=digits(n)); while(#Set(d) == #d, k++; d=digits(k*n)); k;} \\ _Michel Marcus_, Aug 22 2020

%Y Cf. A045538 (where the 2 digits must be consecutive), A337240 (resulting k*n).

%K nonn,base

%O 1,1

%A _Rodolfo Kurchan_, Aug 20 2020