A328567 - OEIS

%I #11 Oct 21 2019 14:45:01

%S 3,6,21,12,75,42,105,24,279,150,341,84,403,210,465,48,1071,558,1197,

%T 300,1323,682,1449,168,1575,806,1701,420,1827,930,1953,96,4191,2142,

%U 4445,1116,4699,2394,4953,600,5207,2646,5461,1364,5715,2898,5969,336,6223,3150

%N 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.

%C This sequence is a binary variant of A309631.

%C 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.

%H Rémy Sigrist, <a href="/A328567/b328567.txt">Table of n, a(n) for n = 1..8192</a>

%H Rémy Sigrist, <a href="/A328567/a328567.gp.txt">PARI program for A328567</a>

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

%e For n = 3:

%e - the first multiples of 3 are (in decimal and in binary), alongside the possible values resulting from striking out a non-trailing zero:

%e     3*k  bin(3*k)  striked (binary)

%e     ---  --------  ----------------

%e       3        11  1

%e       6       110  10

%e       9      1001  1, 100, 101

%e      12      1100  100

%e      15      1111  111

%e      18     10010  10, 1000, 1010

%e      21     10101  101, 1001, 1010, 1011, 1101

%e - 21 is the least appropriate multiple,

%e - so a(3) = 21.

%o (PARI) See Links section.

%Y Cf. A070940, A309631.

%K nonn,base

%O 1,1

%A _Rémy Sigrist_, Oct 20 2019