A328567 a(n) is the smallest positive integer divisible by n such that it is possible to strike out a digit from its binary expansion (apart from trailing zeros) so that the resulting number is nonzero and divisible by n.

3, 6, 21, 12, 75, 42, 105, 24, 279, 150, 341, 84, 403, 210, 465, 48, 1071, 558, 1197, 300, 1323, 682, 1449, 168, 1575, 806, 1701, 420, 1827, 930, 1953, 96, 4191, 2142, 4445, 1116, 4699, 2394, 4953, 600, 5207, 2646, 5461, 1364, 5715, 2898, 5969, 336, 6223, 3150

Offset

1,1

Comments

This sequence is a binary variant of A309631.

This kind of sequence is well defined for any fixed base b > 1: for any n > 0: consider the concatenation in base b, say m, of n, "0", and n; m is a multiple of n, and removing the central "0" (which is not a trailing zero), gives another multiple of n.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..8192

Rémy Sigrist, PARI program for A328567

Formula

Apparently, a(n)/n = 2^(1+A070940(n)) - 1.

Example

For n = 3:
- the first multiples of 3 are (in decimal and in binary), alongside the possible values resulting from striking out a non-trailing zero:
    3*k  bin(3*k)  striked (binary)
    ---  --------  ----------------
      3        11  1
      6       110  10
      9      1001  1, 100, 101
     12      1100  100
     15      1111  111
     18     10010  10, 1000, 1010
     21     10101  101, 1001, 1010, 1011, 1101
- 21 is the least appropriate multiple,
- so a(3) = 21.

Prog

(PARI) See Links section.

Crossrefs

Cf. A070940, A309631.

Sequence in context: A355874 A356967 A370996 * A064366 A260157 A101719

Adjacent sequences: A328564 A328565 A328566 * A328568 A328569 A328570

Keyword

nonn,base

Author

Rémy Sigrist, Oct 20 2019

Status

approved