cp's OEIS Frontend

A116290 Numbers k such that k * (k+6) is the concatenation of a number m with itself.

%I A116290 #8 Apr 08 2025 19:37:53
%S A116290 5,95,462,533,715,819,995,3425,6570,9995,90904,99995,980199,999995,
%T A116290 3636358,6363637,9999995,41176465,58823530,99999995,413533835,
%U A116290 426573427,428571423,432620006,567379989,571428572,573426568,586466160,686261108,725274726,727272722
%N A116290 Numbers k such that k * (k+6) is the concatenation of a number m with itself.
%C A116290 From _Robert Israel_, Apr 08 2025: (Start)
%C A116290 Numbers k such that k * (k + 6) = (10^d + 1) * m for some d and m where m has d digits.
%C A116290 Contains 10^d-5 for all d >= 1. (End)
%H A116290 Robert Israel, <a href="/A116290/b116290.txt">Table of n, a(n) for n = 1..10000</a>
%p A116290 f:= proc(d) local R,t,a,b,x,q;
%p A116290    t:= 10^d+1;
%p A116290    R:= NULL;
%p A116290    for a in numtheory:-divisors(t) do
%p A116290      b:= t/a;
%p A116290      if igcd(a,b) > 1 then next fi;
%p A116290      for x from chrem([0,-6],[a,b]) by t do
%p A116290        q:= x*(x+6)/t;
%p A116290        if q >= 10^d then break fi;
%p A116290        if q >= 10^(d-1) then R:= R, x fi;
%p A116290    od od;
%p A116290    sort([R]);
%p A116290 end proc:
%p A116290 seq(op(f(d)),d=1..10); # _Robert Israel_, Apr 08 2025
%Y A116290 Cf. A116159, A116282, A116289, A116291, A116304.
%K A116290 nonn,base
%O A116290 1,1
%A A116290 _Giovanni Resta_, Feb 06 2006
%E A116290 Name edited by _Robert Israel_, Apr 08 2025