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A057373 Numbers k that can be expressed as k = w + x = y*z with w*x = y^2 + z^2 where w, x, y, and z are all positive integers.

%I A057373 #20 Jan 31 2023 11:56:10
%S A057373 9,18,45,90,117,306,522,585,801,1305,2097,3042,3978,5490,8730,14373,
%T A057373 17730,19485,22698,27234,37629,44109,98514,103338,113013,130365,
%U A057373 155025,186633,257913,290970,405450,602298,675225,884637,1279170,1498185,1767762,1946745
%N A057373 Numbers k that can be expressed as k = w + x = y*z with w*x = y^2 + z^2 where w, x, y, and z are all positive integers.
%C A057373 From _Robert Israel_, Feb 01 2016: (Start)
%C A057373 Numbers k such that k^2 - 4*(d^2 + k^2/d^2) is a square for some divisor d of k.
%C A057373 All terms are divisible by 9.
%C A057373 Includes 9*A001519(k) for all k (where y = 3, z = 3*A001519(k)). In particular, the sequence is infinite. (End)
%p A057373 filter:= proc(n) local x;
%p A057373   nops(select(x -> issqr(n^2-4*x^2 - 4*(n/x)^2), numtheory:-divisors(n)))>0;
%p A057373 end proc:
%p A057373 select(filter, [$1..10^6]); # _Robert Israel_, Feb 01 2016
%t A057373 filterQ[n_] := Length@Select[Divisors[n], IntegerQ@Sqrt[n^2 - 4*#^2 - 4*(n/#)^2]&] > 0;
%t A057373 Select[Range[9, 999999, 9], filterQ] (* _Jean-François Alcover_, Jan 31 2023, after _Robert Israel_ *)
%o A057373 (PARI) is(k) = fordiv(k, y, if(issquare(k^2 - 4*y^2 - 4*sqr(k/y)), return(1))); 0; \\ _Jinyuan Wang_, May 02 2021
%Y A057373 Cf. A001519, A057369, A057370, A057371, A057372, A057444.
%K A057373 nonn
%O A057373 1,1
%A A057373 _Naohiro Nomoto_, Sep 24 2000
%E A057373 a(19)-a(38) from _Robert Israel_, Feb 01 2016
%E A057373 New name from _Jinyuan Wang_, May 02 2021