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