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A116289 Numbers k such that k*(k+5) gives the concatenation of a number m with itself.

%I A116289 #22 Apr 09 2025 07:22:47
%S A116289 6,96,385,429,567,611,814,996,4521,5475,9996,90910,99996,316832,
%T A116289 683164,999996,3636364,6363632,9999996,82352942,99999996,331668332,
%U A116289 368421053,395604391,442767754,461538462,488721800,511278196,538461534,557232242,604395605,631578943,668331664,700089385,727272728
%N A116289 Numbers k such that k*(k+5) gives the concatenation of a number m with itself.
%C A116289 From _Robert Israel_, Apr 09 2025: (Start)
%C A116289 Numbers k such that k * (k + 5) = (10^d + 1) * m for some d and m where m has d digits.
%C A116289 Contains 10^d-4 for all d >= 1. (End)
%H A116289 Robert Israel, <a href="/A116289/b116289.txt">Table of n, a(n) for n = 1..10000</a>
%p A116289 q:= proc(d,m) local R,t,a,b,x,q;
%p A116289    t:= 10^d+1;
%p A116289    R:= NULL;
%p A116289    for a in numtheory:-divisors(t) do
%p A116289      b:= t/a;
%p A116289      if igcd(a,b) > 1 then next fi;
%p A116289      for x from chrem([0,-m],[a,b]) by t do
%p A116289        q:= x*(x+m)/t;
%p A116289        if q >= 10^d then break fi;
%p A116289        if q >= 10^(d-1) then R:= R, x fi;
%p A116289    od od;
%p A116289    sort(convert({R},list));
%p A116289 end proc:
%p A116289 seq(op(q(d,5)),d=1..10); # _Robert Israel_, Apr 09 2025
%Y A116289 Cf. A028557, A020338.
%Y A116289 Cf. A116158, A116281, A116288, A116290, A116303.
%K A116289 nonn,base
%O A116289 1,1
%A A116289 _Giovanni Resta_, Feb 06 2006
%E A116289 More terms from _Robert Israel_, Apr 09 2025