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

%I A383642 #43 Jul 27 2025 18:00:03
%S A383642 2,6,9,12,16,20,28,30,33,34,35,42,48,54,56,58,65,70,72,75,84,86,90,91,
%T A383642 96,105,110,114,120,124,126,128,132,133,152,153,156,160,162,180,182,
%U A383642 189,198,201,205,209,210,217,224,236,238,240,243,246,250,254,258,264,267
%N A383642 Numbers k = x + y with x and y positive integers such that x*y is a cube.
%C A383642 Includes all numbers of the form m*(m + 1).
%C A383642 2 is the only prime member.
%C A383642 If k >= 1 is in the sequence then so is k*m^3 where m >= 1. - _David A. Corneth_, May 05 2025
%e A383642 k=12, 12=3+9, 3*9=3^3.
%e A383642 k=65, 65=25+40, 25*40=10^3.
%t A383642 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]]
%o A383642 (PARI) isok(k) = for(i=1, k\2, if(ispower(i*(k-i), 3), return(1))); \\ _Michel Marcus_, May 04 2025
%o A383642 (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
%o A383642 (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
%Y A383642 Cf. A000578, A337140.
%Y A383642 Supersequence of A003325.
%K A383642 nonn
%O A383642 1,1
%A A383642 _Huaineng He_, May 03 2025