A325775 -id:A325775 - cp's OEIS frontend

A308306 Boomerang numbers: their last digit "comes back" to occupy the place of their first digit (see the Comments section for the explanation).

100, 203, 225, 230, 247, 252, 269, 274, 296, 302, 320, 405, 427, 449, 450, 472, 494, 504, 522, 540, 607, 629, 670, 692, 706, 724, 742, 760, 809, 890, 908, 926, 944, 962, 980, 1012, 1021, 1034, 1043, 1056, 1065, 1078, 1087, 1102, 1120, 1201, 1210, 1223, 1232, 1245, 1254, 1267, 1276, 1289, 1298, 1304, 1322, 1340, 1403, 1425, 1430

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 19 2019

Take 2019; start with 2; jump over 2 cells to the right (as the even digits always move to the right); write 0 on the landing cell; jump over 0 cell to the right (which is the same as moving to the next cell to the right) and write 1 on the landing cell; as 1 is odd, jump over 1 cell to the left; write 9 on the landing cell; jump now over 9 cells to the left and mark A (for "Arrival") on the landing cell. The result will look like this (a dot is a cell): A.......2.901
As this A cell is not the same as the starting one (with "2"), 2019 is not a boomerang number. If we had taken 2011, we would have come back on the starting 2, like this:
2011
2..0
2..01
2.101
A.101
This is why 2011 is in the sequence and 2019 not.
Note that a cell, empty or not, is only a stopover: it can be used several times by different digits.
There are 263499 boomerang numbers < 10^7.
A boomerang number is easy to find, knowing the hereunder definition:
Integers B such that (the number of even digits + the sum of those) = (the number of odd digits + the sum of those).
Note: this sequence is not related to A256174 ("Boomerang fractions").

			7308403 is a boomerang number as we have 4 even digits with sum 12 (4+12=16) and 3 odd digits with sum 13 (3+13=16).

Jean-Marc Falcoz, Table of n, a(n) for n = 1..28444

CF. A325775 and A325776 which play with the same concept.

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.

Original entry on oeis.org

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

Eric Angelini and Jean-Marc Falcoz, May 20 2019

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.

			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

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

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

New title from Charlie Neder, Jun 03 2019

cp's OEIS Frontend

A308306 Boomerang numbers: their last digit "comes back" to occupy the place of their first digit (see the Comments section for the explanation).

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