%I #45 Nov 09 2024 19:31:56

%S 3,2,7,6,5,308,51,34217,7794416

%N Smallest nontrivial number m that when expressed in base n and then interpreted in base 10, is a multiple of the original number.

%C Not defined for n>9.

%C Note that for n=1, the unitary numeral system is used, whereas for n>=2, it is the usual base-dependent positional numeral system. - _Michel Marcus_, Sep 16 2020

%e a(1)=3 since 3 in base 1 ('unary') is 111 and, interpreted in base 10, 111 is a number divisible by 3, whereas 2 in base 1 is 11 and, interpreted in base 10, 11 is not divisible by 2.

%e a(3)=7 since 7 in base 3 is 21 and, interpreted in base 10, 21 is divisible by 7, whereas 3, 4, 5, 6 in base 3 are, respectively, 10, 11, 12, 20 and, interpreted in base 10, these are numbers not divisible, respectively, by 3, 4, 5, 6.

%e Trivial solutions (i.e., m<n for n>1 and m=n for n=1) are not considered here.

%o (Python)

%o m,M=2,11

%o while M%m>0:

%o m,M=m+1,10*M+1

%o print(1,m)

%o for n in range(2,10):

%o m,M,d=n,10,10-n

%o while M%m>0:

%o m=m+1

%o u,s=m,0

%o while u%n==0:

%o u,s=u//n,10*s+d

%o M=M+1+s

%o print(n,m)

%o # _Dimiter Skordev_, Sep 28 2020

%Y Cf. A032551, A032551, A032532.

%K base,fini,full,nonn,changed

%O 1,1

%A _Sergio Pimentel_, Jun 06 2008

%E a(9) corrected by _Giovanni Resta_, Nov 02 2017