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%I #14 Jul 15 2018 14:00:49
%S 1,2,4,12,13,15,16,17,20,24,26,28,32,35,36,49,60,66,70,75,76,100,105,
%T 112,119,120,121,128,132,136,149,156,159,160,164,165,166,167,176,186,
%U 207,215,235,258,266,268,344,352,356,391,416,429,499,610,625,671,675,690,695,750
%N 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.
%H Jean-Marc Falcoz, <a href="/A316634/b316634.txt">Table of n, a(n) for n = 1..1352</a>
%e 1 is in the sequence because 12/(1+2) is the integer 4;
%e 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);
%e 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;
%e 4 is in the sequence because 45/(4+5) is the integer 5;
%e 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;
%e . . .
%e 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.
%e . . .
%e 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.
%e Etc.
%Y Cf. A316534 and A316538 for other ways to concatenate/divide and iterate the procedure.
%K base,nonn
%O 1,2
%A _Eric Angelini_ and _Jean-Marc Falcoz_, Jul 09 2018
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