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.

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

Offset: 1

Author

Eric Angelini and Jean-Marc Falcoz, May 20 2019

Keywords

• sign
• base

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.

Examples

			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
		

Links

• Jean-Marc Falcoz, Table of n, a(n) for n = 1..12000

Crossrefs

Cf. A308306 (the "boomerang numbers"), and A325775 (where duplicate terms are not admitted).

Extensions

New title from Charlie Neder, Jun 03 2019