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A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.

%I A116286 #19 Apr 09 2025 07:22:12
%S A116286 9,99,427,572,726,845,999,7809,9999,36364,63635,99999,326733,673266,
%T A116286 999999,4545453,5454546,9999999,47058822,52941177,99999999,331983806,
%U A116286 332667333,384615385,422892897,475524476,524475523,577107102,615384614,667332666,668016193,719964245,758241757,804511279
%N A116286 Numbers k such that k*(k+2) gives the concatenation of a number m with itself.
%C A116286 From _Robert Israel_, Apr 08 2025: (Start)
%C A116286 Numbers k such that k * (k + 2) = (10^d + 1) * m for some d and m where m has d digits.
%C A116286 Contains 10^d - 1 for all d >= 1. (End)
%H A116286 Robert Israel, <a href="/A116286/b116286.txt">Table of n, a(n) for n = 1..10000</a>
%p A116286 q:= proc(d,m) local R,t,a,b,x,q;
%p A116286    t:= 10^d+1;
%p A116286    R:= NULL;
%p A116286    for a in numtheory:-divisors(t) do
%p A116286      b:= t/a;
%p A116286      if igcd(a,b) > 1 then next fi;
%p A116286      for x from chrem([0,-m],[a,b]) by t do
%p A116286        q:= x*(x+m)/t;
%p A116286        if q >= 10^d then break fi;
%p A116286        if q >= 10^(d-1) then R:= R, x fi;
%p A116286    od od;
%p A116286    sort(convert({R},list));
%p A116286 end proc:
%p A116286 A:=[seq(op(q(d,2)),d=1..10)]; # _Robert Israel_, Apr 08 2025
%Y A116286 Cf. A116136, A116279, A116285, A116287, A116295.
%K A116286 nonn,base
%O A116286 1,1
%A A116286 _Giovanni Resta_, Feb 06 2006
%E A116286 More terms from _Robert Israel_, Apr 08 2025