A139285 Smallest nontrivial number m that when expressed in base n and then interpreted in base 10, is a multiple of the original number.

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

Offset: 1

Author

Sergio Pimentel, Jun 06 2008

Keywords

• base
• fini
• full
• nonn

Comments

Not defined for n>9.

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

Examples

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

Crossrefs

Cf. A032551, A032551, A032532.

Programs

Python

m,M=2,11
while M%m>0:
    m,M=m+1,10*M+1
print(1,m)
for n in range(2,10):
    m,M,d=n,10,10-n
    while M%m>0:
        m=m+1
        u,s=m,0
        while u%n==0:
            u,s=u//n,10*s+d
        M=M+1+s
    print(n,m)
# Dimiter Skordev, Sep 28 2020

Extensions

a(9) corrected by Giovanni Resta, Nov 02 2017