A359225 Numbers that can be expressed as (a^3 + b^3)/(a*b) with b > a >= 1.

9, 18, 27, 28, 35, 36, 45, 54, 56, 63, 65, 70, 72, 81, 84, 90, 91, 99, 105, 108, 112, 117, 126, 130, 133, 135, 140, 144, 152, 153, 162, 168, 171, 175, 180, 182, 189, 195, 196, 198, 207, 210, 216, 217, 224, 225, 234, 243, 245, 252, 260, 261, 266, 270, 273, 279, 280, 288, 297

Offset: 1

Author

Zhining Yang, Dec 22 2022

Keywords

• nonn
• easy

Comments

Numbers k such that k*a*b = a^3 + b^3 has integer solutions with b > a >= 1.

Numbers of the form r*(s^3 + t^3) with r >= 1 and s > t >= 1, by a = r*s*t^2, b = r*s^2*t.

Examples

			63 can be expressed as (14^3 + 28^3)/(14*28) so 63 is a term.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..22000

Crossrefs

Cf. A009003, A024670 (subsequence), A373973 (characteristic function).

Positions of positive terms in A373974.

Programs

MATLAB

function a = A359225( max_n )
    OneToN = [1:max_n]; a = [];
    for n = 1:max_n-1
        A = (OneToN(1:n)'*ones(1,max_n-n)).^3 ...
          + (ones(n,1)*OneToN(n+1:end)).^3;
        a = unique([a reshape(A(:),1,numel(A))]);
        a = a(1:min(length(a),max_n));
    end
    A = a'*OneToN;
    a = unique(A(:)); a = a(1:min(length(a),max_n))';
end

Mathematica

n = 300; Union@
 Sort@Flatten@
   Table[r*(s^3 + t^3), {r, 1, n/9}, {s, 1,
     CubeRoot[n/(2*r) - 1]}, {t, s + 1, CubeRoot[n/r - s^3]}]

PARI

isA359225 = A373973; \\ Antti Karttunen, Jun 24 2024

Python

def aupto(limit):
    c=[k**3 for k in range(1,limit) if k**3<=limit]
    s=set()
    for i in range(len(c)):
        for j in range(i+1,len(c)):
            t=(c[i]+c[j])
            for r in range(1, limit//t+1) :
                s.add(r*t)
    return(sorted(s))
print(aupto(500))