A325776 -id:A325776 - 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.

A325775 Numbers that bring back the last digit of n on its first digit, using the "boomerang protocol" explained in A308306, with no duplicate term (except the -1 terms). See the Comments and Example sections for details.

Original entry on oeis.org

102, 25, 20, 27, 22, 29, 24, 90, 26, 0, 104, 10, 40, 30, 42, 49, 44, 70, 28, 3, 23, 5, 12, 7, 2, 9, 4, 19, 6, 14, 106, 21, 60, 32, 62, 47, 46, 50, 48, 13, 52, 15, 45, 17, 34, 37, 36, 39, 16, 38, 108, 41, 80, 43, 64, 54, 66, 69, 68, 33, 72, 35, 74, 55, 67, 57, 56, 59, 58, 18, 128, 61, 82, 63, 84, 65, 86, 76, 88, 53, 92, 73, 94, 75, 96

Offset: 1

Eric Angelini and Jean-Marc Falcoz, May 20 2019

All the nonnegative integers will appear in this sequence, except the terms of A308306.
The "boomerang protocol" sends 1 to the left (as 1 is odd - the even digits move to the right), 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 for instance 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 is the Arrival cell we must 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....
We see that 1 jumps to the left over 1 cell, 0 to the right over 0 cell (thus moving to this cell), 2 jumps over 2 cells and lands precisely on the Arrival cell.
Note that many integers can "bring back" 1 in its initial cell, 120 is one of them, for instance, or 1410.

			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 formula being that a(n) brings 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.

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

Cf. A308306 (the "boomerang numbers") and A325776 where duplicate terms are admitted.

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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