OFFSET
1,1
COMMENTS
Old title was: "a(n) is the smallest number bringing back n on its first digit, using the 'boomerang protocol' explained in A308306."
A similar sequence, but with no duplicate term, is A325775.
If a(n) = -1, then n is a "boomerang number" (see A308306).
The "boomerang protocol" sends 1 to the left (as 1 is odd), jumping over exactly 1 cell. To "bring back" 1 to its initial cell, the smallest integer is 102. Let's see how:
Our initial 1 starts here (dots are cells):
....1....
and ends there (S is the starting cell):
..1.S....
We have this pattern now for the "bring back" integer (S is the new start, A the Arrival cell we want to reach, which was the starting cell of 1):
..S.A....
The smallest integer starting on S and ending on A is 102:
..1.A....
0...A....
.2..A....
The integer 120 does the same job, as does also 1410 - but we keep 102 as 102 is the smallest available integer.
LINKS
Jean-Marc Falcoz, Table of n, a(n) for n = 1..12000
EXAMPLE
The sequence starts with 102,25,20,27,22,29,24,90,... We see that:
a(1) = 102 means that 102 will bring 1 back in its initial cell;
a(2) = 25 means that 25 will bring 2 back in its initial cell;
a(3) = 20 means that 20 will bring 3 back in its initial cell;
a(4) = 27 means that 27 will bring 4 back in its initial cell;
a(5) = 22 means that 22 will bring 5 back in its initial cell;
The general rule being that a(n) is the smallest number bringing back n in its initial cell.
a(100) = -1 means that 100 is a "boomerang number": it "comes back" by itself without any external help. Those numbers are listed in A308306.
a(1) is not 0, since even though the least boomerang number beginning with 1 is 100, leading zeroes are not allowed. - Charlie Neder, Jun 03 2019
CROSSREFS
KEYWORD
sign,base
AUTHOR
Eric Angelini and Jean-Marc Falcoz, May 20 2019
EXTENSIONS
New title from Charlie Neder, Jun 03 2019
STATUS
approved