A316634 Numbers k such that k concatenated with k+1 and then divided by 2k+1 produces an integer after some divisions explained in the Example section.

1, 2, 4, 12, 13, 15, 16, 17, 20, 24, 26, 28, 32, 35, 36, 49, 60, 66, 70, 75, 76, 100, 105, 112, 119, 120, 121, 128, 132, 136, 149, 156, 159, 160, 164, 165, 166, 167, 176, 186, 207, 215, 235, 258, 266, 268, 344, 352, 356, 391, 416, 429, 499, 610, 625, 671, 675, 690, 695, 750

Offset

1,2

Links

Jean-Marc Falcoz, Table of n, a(n) for n = 1..1352

Example

1 is in the sequence because 12/(1+2) is the integer 4;
2 is in the sequence though 23/(2+3) is not an integer, because if we do floor(23/(2+3)) we get 4, and if we use this 4 to do now 23/(2+4) we get an integer (here 4 again);
3 is not in the sequence because 34/(3+4) is not an integer and even if we try floor(34/(3+4)) = 3, we will be stuck in the loop 34/(3+4) =. . . never ending in an integer;
4 is in the sequence because 45/(4+5) is the integer 5;
5 is not in the sequence because 56/(5+6) is not an integer and even if we try floor(56/(5+6))=5, we will be stuck in the loop 56/(5+6) =. . . never ending in an integer;
. . .
13 is in the sequence though 1314/(13+14) is not an integer, but if we repeatedly apply the "floor" trick, we will produce an integer at the end (here 34): floor(1314/(13+14))= 48, then floor(1348/(13+48)) = 22, then floor(1322/(13+22)) = 37, then floor(1337/(13+37)) = 26 and finally 1326/(13+26) produces the integer 34, without the help of the "floor" function.
. . .
761 is not in the sequence because 761762/(761+762) is not an integer and even if we repeatedly apply the "floor" trick, we will be stuck in a loop: floor(761500/(761+500)) = 603, then floor(761603/(761+603)) = 558, then floor(762558/(761+558)) = 577, then floor(761577/(761+577)) = 569, then floor(761569(761+569)) = 572, then floor(761572/(761+572)) = 571, then floor(761571/(761+571)) = 571 again.
  Etc.

Crossrefs

Cf. A316534 and A316538 for other ways to concatenate/divide and iterate the procedure.

Sequence in context: A087796 A268494 A306851 * A039587 A194368 A111069

Adjacent sequences: A316631 A316632 A316633 * A316635 A316636 A316637

Keyword

base,nonn

Author

Eric Angelini and Jean-Marc Falcoz, Jul 09 2018

Status

approved