

A325776


a(n) is the least nonnegative integer such that n concatenated with a(n) is a boomerang number (A308306), or 1 if n is a boomerang number.


3



102, 25, 20, 27, 22, 29, 24, 90, 26, 0, 20, 10, 22, 25, 24, 27, 26, 29, 28, 3, 10, 5, 0, 7, 2, 9, 4, 19, 6, 2, 22, 0, 24, 10, 26, 25, 28, 27, 48, 5, 25, 7, 10, 9, 0, 19, 2, 39, 4, 4, 24, 2, 26, 0, 28, 10, 48, 25, 68, 7, 27, 9, 25, 19, 10, 39, 0, 59, 2, 6, 26, 4, 28, 2, 48, 0, 68, 10, 88, 9, 29, 19, 27, 39, 25, 59, 10, 79, 0, 8, 28, 6
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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

JeanMarc 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

Cf. A308306 (the "boomerang numbers"), and A325775 (where duplicate terms are not admitted).
Sequence in context: A006064 A230304 A015164 * A325775 A204749 A244949
Adjacent sequences: A325773 A325774 A325775 * A325777 A325778 A325779


KEYWORD

sign,base


AUTHOR

Eric Angelini and JeanMarc Falcoz, May 20 2019


EXTENSIONS

New title from Charlie Neder, Jun 03 2019


STATUS

approved



