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

1,1

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)

Links

Table of n, a(n) for n=1..38.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A138900 A202187 A370016 * A153185 A325450 A212345

Adjacent sequences: A057370 A057371 A057372 * A057374 A057375 A057376

Keyword

nonn

Author

Naohiro Nomoto, Sep 24 2000

Extensions

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

New name from Jinyuan Wang, May 02 2021

Status

approved