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Squares arising as a concatenation of k and 9's complement of k.
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%I #16 Sep 10 2020 03:01:54

%S 36,81,1089,4356,9801,110889,443556,998001,11108889,44435556,99980001,

%T 1111088889,4444355556,9999800001,111110888889,444443555556,

%U 999998000001,11111108888889,44444435555556,99999980000001

%N Squares arising as a concatenation of k and 9's complement of k.

%C From _Robert Israel_, Sep 09 2020: (Start)

%C Numbers of the form j^2*x*(10^k-1) where x = A007913(10^k-1) and 10^(k-1)+1 <= j^2*x <= 10^k-1.

%C If k >= 2 is not in A046412, there are only three terms with 2*k digits, namely (10^k-1)^2/9, 4*(10^k-1)^2/9, and 9*(10^k-1)^2/9.

%C The first term not of one of those three forms is a(25)=197530863802469136.

%C (End)

%H Robert Israel, <a href="/A084006/b084006.txt">Table of n, a(n) for n = 1..367</a>

%e 1089 = 33^2 is a concatenation of 10 and 89, 10+89 = 99.

%p f:= proc(k) local F,x,p,t;

%p p:= 10^k-1;

%p F:= select(t -> t[2]::odd, ifactors(p)[2]);

%p x:= mul(t[1],t=F);

%p seq(j^2*x*p, j=ceil(sqrt((10^(k-1)+1)/x))..floor(sqrt(p/x)))

%p end proc:

%p map(f, [$1..20]); # _Robert Israel_, Sep 09 2020

%Y Cf. A007913, A046412, A084004, A084005, A084007.

%K base,nonn

%O 1,1

%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

%E More terms from _Ray Chandler_, May 31 2003

%E Offset changed by _Robert Israel_, Sep 09 2020