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

Author

Rémy Sigrist, Oct 20 2019

Keywords

• nonn
• base

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.

Examples

			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.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..8192
• Rémy Sigrist, PARI program for A328567

Crossrefs

Cf. A070940, A309631.

Programs

PARI

See Links section.

Formula

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