A370571 Smallest multiple of n that when written in base 10 uses only 0's and 1's and at least one of each.

10, 10, 1011, 100, 10, 1110, 1001, 1000, 1011111111, 10, 110, 11100, 1001, 10010, 1110, 10000, 11101, 1111111110, 11001, 100, 10101, 110, 110101, 111000, 100, 10010, 1101111111, 100100, 1101101, 1110, 111011, 100000, 1101111, 111010, 10010, 11111111100, 1110, 110010, 10101, 1000

Offset: 1

Author

Ivan N. Ianakiev, Feb 22 2024

Keywords

• nonn
• base

Comments

For all n, a(n) exists (see proof in References).

References

• Peter Winkler, Mathematical Puzzles (revised edition), CRC Press, 2024, p. liii.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A004290, A244859, A007088.

Programs

Mathematica

a[n_]:=Min[Select[FromDigits/@Tuples[{0,1},n+1],
Divisible[#,n]&&Union[IntegerDigits[#]]=={0,1}&]]; a/@Range[23]

Python

from itertools import count
def a(n): return next(d for k in count(1) if ("0" in (b:=bin(k)[2:])) and (d:=int(b))%n==0)
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Feb 22 2024

Formula

a(10^e-1) <= 1^e 0 1^(8*e), where ^ denotes repeated concatenation of digits on the right-hand side. - Michael S. Branicky, Feb 22 2024

Extensions

More terms from Michael S. Branicky, Feb 22 2024