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A116292 Numbers k such that k * (k + 8) is the concatenation of a number m with itself.

%I A116292 #9 Jul 05 2025 18:33:30
%S A116292 3,93,377,616,707,902,993,8760,9993,45455,54538,99993,693062,999993,
%T A116292 8181811,9999993,88235287,99999993,327935223,330669331,363636364,
%U A116292 418318516,428571429,461538454,538461539,571428564,581681477,636363629,669330662,672064770,691571587,756506652,781954880,789473685
%N A116292 Numbers k such that k * (k + 8) is the concatenation of a number m with itself.
%C A116292 From _Robert Israel_, Apr 09 2025: (Start)
%C A116292 Numbers k such that k * (k + 6) = (10^d + 1) * m for some d and m where m has d digits.
%C A116292 Contains 10^d-7 for all d >= 1. (End)
%H A116292 Robert Israel, <a href="/A116292/b116292.txt">Table of n, a(n) for n = 1..10000</a>
%p A116292 q:= proc(d,m) local R,t,a,b,x,q;
%p A116292    t:= 10^d+1;
%p A116292    R:= NULL;
%p A116292    for a in numtheory:-divisors(t) do
%p A116292      b:= t/a;
%p A116292      if igcd(a,b) > 1 then next fi;
%p A116292      for x from chrem([0,-m],[a,b]) by t do
%p A116292        q:= x*(x+m)/t;
%p A116292        if q >= 10^d then break fi;
%p A116292        if q >= 10^(d-1) then R:= R, x fi;
%p A116292    od od;
%p A116292    sort(convert({R},list));
%p A116292 end proc:
%p A116292 seq(op(q(d,8)),d=1..10); # _Robert Israel_, Apr 09 2025
%t A116292 ccnQ[n_]:=With[{ccc=With[{c=n(n+8)},TakeDrop[IntegerDigits[c],IntegerLength[c]/2]]},ccc[[1]]==ccc[[2]]];  Select[Range[10^6],ccnQ]//Quiet (* The program generates the first 14 terms of the sequence. *) (* _Harvey P. Dale_, Jul 05 2025 *)
%Y A116292 Cf. A116161, A116277, A116291, A116293, A116299.
%K A116292 nonn,base
%O A116292 1,1
%A A116292 _Giovanni Resta_, Feb 06 2006
%E A116292 Name edited and more terms from _Robert Israel_, Apr 09 2025