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.

9, 18, 45, 90, 117, 306, 522, 585, 801, 1305, 2097, 3042, 3978, 5490, 8730, 14373, 17730, 19485, 22698, 27234, 37629, 44109, 98514, 103338, 113013, 130365, 155025, 186633, 257913, 290970, 405450, 602298, 675225, 884637, 1279170, 1498185, 1767762, 1946745

Offset: 1

Author

Naohiro Nomoto, Sep 24 2000

Keywords

• nonn

Comments

From Robert Israel, Feb 01 2016: (Start)

Numbers k such that k^2 - 4*(d^2 + k^2/d^2) is a square for some divisor d of k.

All terms are divisible by 9.

Includes 9*A001519(k) for all k (where y = 3, z = 3*A001519(k)). In particular, the sequence is infinite. (End)

Crossrefs

Cf. A001519, A057369, A057370, A057371, A057372, A057444.

Programs

Maple

filter:= proc(n) local x;
  nops(select(x -> issqr(n^2-4*x^2 - 4*(n/x)^2), numtheory:-divisors(n)))>0;
end proc:
select(filter, [$1..10^6]); # Robert Israel, Feb 01 2016

Mathematica

filterQ[n_] := Length@Select[Divisors[n], IntegerQ@Sqrt[n^2 - 4*#^2 - 4*(n/#)^2]&] > 0;
Select[Range[9, 999999, 9], filterQ] (* Jean-François Alcover, Jan 31 2023, after Robert Israel *)

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

Extensions

a(19)-a(38) from Robert Israel, Feb 01 2016

New name from Jinyuan Wang, May 02 2021