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

1,1

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

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A004290, A244859, A007088.

Sequence in context: A288504 A287754 A287495 * A288337 A288806 A279716

Adjacent sequences: A370568 A370569 A370570 * A370572 A370573 A370574

Keyword

nonn,base

Author

Ivan N. Ianakiev, Feb 22 2024

Extensions

More terms from Michael S. Branicky, Feb 22 2024

Status

approved