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A115529 Numbers k such that the concatenation of 2*k with k gives a square.

%I A115529 #9 Jul 15 2021 06:28:04
%S A115529 1632653061224489796,3673469387755102041,6530612244897959184,
%T A115529 108166576527852893455922120064900,130881557598702001081665765278529,
%U A115529 155759870200108166576527852893456,182801514332071389940508382909681
%N A115529 Numbers k such that the concatenation of 2*k with k gives a square.
%C A115529 Numbers of the form k = a*b^2 where 10^(d-1) <= k < 10^d and (2*10^d+1)/a is a square. - _Robert Israel_, Jan 13 2021
%H A115529 Robert Israel, <a href="/A115529/b115529.txt">Table of n, a(n) for n = 1..36</a>
%e A115529 3265306122448979592_1632653061224489796 = 5714285714285714286^2.
%p A115529 f:= proc(d) local R,q,F,G,s,t,a,u,i;
%p A115529    q:= 2*10^d+1;
%p A115529    F:= ifactors(q)[2];
%p A115529    G:= map(t -> [t[1],floor(t[2]/2)], F);
%p A115529    s:= mul(t[1]^t[2],t=G);
%p A115529    R:= NULL:
%p A115529    for a in numtheory:-divisors(s) do
%p A115529      u:= q/a^2;
%p A115529      R:= R, seq(i^2*u, i=ceil(sqrt(10^(d-1)/u))..floor(sqrt((10^d-1)/u)))
%p A115529    od;
%p A115529    R
%p A115529 end proc:
%p A115529 seq(f(d),d=1..33); # _Robert Israel_, Jan 13 2021
%Y A115529 Cf. A102567, A106497, A115527 - A115556.
%K A115529 nonn,base
%O A115529 1,1
%A A115529 _Giovanni Resta_, Jan 25 2006