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%I #24 Sep 14 2023 00:47:07
%S 65,6653,9605,218413,283720,996005,58446925,99960005,6086712229,
%T 7385370133,8478948853,9999600005,120178240093,161171620229,
%U 358247912200,426843573160,893417179213,999996000005,23376713203604
%N Numbers k such that the concatenation of k with k-4 gives a square.
%C The terms of this sequence (k//k-4 = m*m), A116104 (k//k-8 = m*(m+4)) and A116121 (k//k-5 = m*(m+2)) agree as long as the two concatenated numbers k and k-x have the same length. This condition is satisfied for the given terms of all three sequences. - _Georg Fischer_, Sep 12 2022
%C From _Robert Israel_, Sep 13 2023: (Start)
%C Numbers k of the form (y^2+4)/(10^d + 1) where 10^(d-1) <= k - 4 < 10^d and y is a square root of -4 mod (10^d + 1).
%C Includes 10^(2*d) - 4*10^d + 5 for all d >= 1, as the concatenation of this with 10^(2*d) - 4*10^d + 1 is 10^(4*d) - 4 * 10^(3*d) + 6 * 10^(2*d) - 4 * 10^d + 1 = (10^d - 1)^4.
%C This is the same sequence as A116104 and A116121. The only possible differences would be if 10^(d-1) + 4 <= k <= 10^(d-1) + 7 or 10^d + 4 <= k <= 10^d + 7, so that k - 4 and k - 8 have different numbers of digits.
%C But in none of those cases can (10^d + 1)*k - 4 be a square:
%C If k = 10^(d-1) + 4 or 10^d + 4, (10^d + 1)*k - 4 == 6 (mod 9).
%C If k = 10^(d-1) + 5 or 10^d + 5, (10^d + 1)*k - 4 == 2 (mod 3).
%C If k = 10^(d-1) + 6 or 10^d + 6, (10^d + 1)*k - 4 == 2 (mod 10).
%C If k = 10^(d-1) + 7 or 10^d + 7, (10^d + 1)*k - 4 == 3 (mod 10). (End)
%H Robert Israel, <a href="/A115432/b115432.txt">Table of n, a(n) for n = 1..2805</a>
%e 9605_9601 = 9801^2.
%p f:= proc(d) uses NumberTheory; local m,r;
%p m:= 10^d + 1;
%p if QuadraticResidue(-4,m) = -1 then return NULL fi;
%p r:= ModularSquareRoot(-4, m);
%p op(sort(select(t -> t >= 10^(d-1)+4 and t < 10^d+4, map(t -> ((r*t mod m)^2+4)/m, convert(RootsOfUnity(2,m),list)))))
%p end proc:
%p map(f, [$1..20]); # _Robert Israel_, Sep 12 2023
%Y Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115431, A115433, A115434, A115435, A115436, A115443.
%Y Almost the same as A116104 and A116121.
%K base,nonn
%O 1,1
%A _Giovanni Resta_, Jan 25 2006
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