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A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.

%I A116285 #23 Jul 11 2025 08:46:56
%S A116285 363,637,714,923,8905,81818,336633,663367,7272727,76470589,333666333,
%T A116285 405436668,428571429,447710185,454545454,473684211,526315789,
%U A116285 545454546,552289815,571428571,594563332,666333667,692307693,711446449,762237762,834008097,859982123,879120879,902255640,974025975,980861244
%N A116285 Numbers k such that k * (k+1) is the concatenation of a number m with itself.
%C A116285 From _Robert Israel_, Apr 08 2025: (Start)
%C A116285 Numbers k such that k * (k + 1) = (10^d + 1) * m for some d and m where m has d digits.
%C A116285 Includes (10^(3*d)-1)/3 + (10^d-1)*10^d/3 and 2*(10^(3*d)-1)/3 - (10^d-1)*10^d/3 + 1 for all d >= 1. (End)
%H A116285 Robert Israel, <a href="/A116285/b116285.txt">Table of n, a(n) for n = 1..10000</a>
%F A116285 A161356(n) = a(n)*(a(n)+1). - _Michael S. Branicky_, Jul 11 2025
%p A116285 q:= proc(d,m) local R,t,a,b,x,q;
%p A116285    t:= 10^d+1;
%p A116285    R:= NULL;
%p A116285    for a in numtheory:-divisors(t) do
%p A116285      b:= t/a;
%p A116285      if igcd(a,b) > 1 then next fi;
%p A116285      for x from chrem([0,-m],[a,b]) by t do
%p A116285        q:= x*(x+m)/t;
%p A116285        if q >= 10^d then break fi;
%p A116285        if q >= 10^(d-1) then R:= R, x fi;
%p A116285    od od;
%p A116285    sort(convert({R},list));
%p A116285 end proc:
%p A116285 seq(op(q(d,1)),d=1..10); # _Robert Israel_, Apr 08 2025
%Y A116285 Cf. A116154, A116272, A116286, A116294, A161356.
%K A116285 nonn,base
%O A116285 1,1
%A A116285 _Giovanni Resta_, Feb 06 2006
%E A116285 More terms from _Robert Israel_, Apr 08 2025