A383642 Numbers k = x + y with x and y positive integers such that x*y is a cube.

2, 6, 9, 12, 16, 20, 28, 30, 33, 34, 35, 42, 48, 54, 56, 58, 65, 70, 72, 75, 84, 86, 90, 91, 96, 105, 110, 114, 120, 124, 126, 128, 132, 133, 152, 153, 156, 160, 162, 180, 182, 189, 198, 201, 205, 209, 210, 217, 224, 236, 238, 240, 243, 246, 250, 254, 258, 264, 267

Offset: 1

Author

Huaineng He, May 03 2025

Keywords

• nonn

Comments

Includes all numbers of the form m*(m + 1).

2 is the only prime member.

If k >= 1 is in the sequence then so is k*m^3 where m >= 1. - David A. Corneth, May 05 2025

Examples

			k=12, 12=3+9, 3*9=3^3.
k=65, 65=25+40, 25*40=10^3.
		

Crossrefs

Cf. A000578, A337140.

Supersequence of A003325.

Programs

Mathematica

kMax = 300; result = {}; For[k = 2, k <= kMax, k++, For[a = 1, a < k, a++, b = k - a; product = a * b; cubeRoot = Round[product^(1 / 3)]; If[cubeRoot^3 == product, result = Append[result, k];]]]; Sort[Union[result]]

PARI

isok(k) = for(i=1, k\2, if(ispower(i*(k-i), 3), return(1))); \\ Michel Marcus, May 04 2025

PARI

is(n) = {my(maxc = sqrtnint(((n/2)^2)\1, 3)); for(i = 1, maxc, if(issquare(n^2 - 4*i^3, &sqrtD), P = (n + sqrtD)/2; if(denominator(P) == 1, return(1)))); 0} \\ David A. Corneth, May 05 2025

PARI

upto(n) = {my(maxc = sqrtnint(((n/2)^2)\1, 3), res = List(), f); for(i = 1, maxc, f = factor(i); f[,2]*=3; d = divisors(f); forstep(j = (#d+1)\2, 1, -1, c = d[j] + d[#d + 1 - j]; if(c <= n, listput(res, c), next(2)))); Set(res)} \\ David A. Corneth, May 05 2025