A207778 Smallest multiple of 2^n using only 1's and 2's.

1, 2, 12, 112, 112, 2112, 2112, 122112, 122112, 12122112, 12122112, 12122112, 111212122112, 1111212122112, 11111212122112, 11111212122112, 11111212122112, 11111212122112, 111211111212122112, 111211111212122112, 111211111212122112, 111211111212122112

Offset

0,2

Comments

An induction-based argument can be used to show that this sequence is actually infinite.

Problem 1, proposed during the 5th All-Soviet-Union Mathematical Competition in 1971 at Riga (Pertsel link), asks for a proof that this sequence is infinite. - Bernard Schott, Mar 20 2023

References

J. B. Tabov and P. J. Taylor, Methods of Problem Solving, Book 1, Australian Mathematics Trust, 1996.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Vladimir A. Pertsel, Problems of the All-Soviet-Union Mathematical Competitions 1961-1986, the 5th competition, Riga, 1971, problem 144.

Index to sequences related to Olympiads and other Mathematical competitions.

Formula

a(n) <= A053312(n).

Mathematica

Table[m = 1; p = 2^k; While[Total@ DigitCount[m p][[3 ;; -1]] > 0, m++]; m p, {k, 0, 11}] (* Michael De Vlieger, Mar 17 2023 *)

Prog

(PARI) a(n) = my(k=1, d=digits(k*2^n)); while (!((vecmin(d)>=1) && (vecmax(d)<=2)), k++; d=digits(k*2^n)); k*2^n; \\ Michel Marcus, Mar 15 2023

Crossrefs

Cf. A000079, A007931, A023396, A053312, A126933.

Sequence in context: A296644 A235860 A317208 * A102659 A212659 A191895

Adjacent sequences: A207775 A207776 A207777 * A207779 A207780 A207781

Keyword

nonn,base

Author

Lekraj Beedassy, Feb 20 2012

Extensions

More terms from Alois P. Heinz, Feb 20 2012

Status

approved