cp's OEIS Frontend

A116291 Numbers k such that k * (k + 7) is the concatenation of a number m with itself.

%I A116291 #7 Apr 09 2025 07:22:00
%S A116291 4,94,455,539,994,7665,9994,72721,99994,356429,643565,999994,9090910,
%T A116291 9999994,35294118,64705876,99999994,335664329,664335665,684210520,
%U A116291 818181819,838056674,846153847,866028702,980125139,999999994
%N A116291 Numbers k such that k * (k + 7) is the concatenation of a number m with itself.
%C A116291 From _Robert Israel_, Apr 09 2025: (Start)
%C A116291 Numbers k such that k * (k + 7) = (10^d + 1) * m for some d and m where m has d digits.
%C A116291 Contains 10^d - 6 for all d >= 1. (End)
%H A116291 Robert Israel, <a href="/A116291/b116291.txt">Table of n, a(n) for n = 1..10000</a>
%p A116291 q:= proc(d,m) local R,t,a,b,x,q;
%p A116291    t:= 10^d+1;
%p A116291    R:= NULL;
%p A116291    for a in numtheory:-divisors(t) do
%p A116291      b:= t/a;
%p A116291      if igcd(a,b) > 1 then next fi;
%p A116291      for x from chrem([0,-m],[a,b]) by t do
%p A116291        q:= x*(x+m)/t;
%p A116291        if q >= 10^d then break fi;
%p A116291        if q >= 10^(d-1) then R:= R, x fi;
%p A116291    od od;
%p A116291    sort(convert({R},list));
%p A116291 end proc:
%p A116291 [seq(op(q(d,7)),d=1..10)]; # _Robert Israel_, Apr 09 2025
%Y A116291 Cf. A116160, A116283, A116290, A116292, A116298.
%K A116291 nonn,base
%O A116291 1,1
%A A116291 _Giovanni Resta_, Feb 06 2006