A268319 -id:A268319 - cp's OEIS frontend

A186885 Numbers whose squares are the average of two distinct positive cubes.

6, 42, 48, 78, 147, 162, 196, 336, 384, 456, 624, 722, 750, 1050, 1134, 1176, 1296, 1342, 1568, 1573, 1674, 1694, 2028, 2058, 2106, 2366, 2387, 2450, 2522, 2646, 2688, 2899, 3072, 3087, 3211, 3648, 3698, 3969, 4374, 4992, 5250, 5292, 5550, 5776, 5915, 6000

Offset: 1

Zak Seidov, Feb 28 2011

nonn

If m is in this sequence, then so is m*k^3 for all k >= 1: e.g., both m = 6 and 6000 = m*10^3 are in this sequence. Also, there are no primes in this sequence.
The table gives all 396 triples (n, a, b) such that n^2 = (a^3 + b^3)/2 and n < 5*10^5.
Parities of a and b are equal: a == b (mod 2). - David A. Corneth, Oct 13 2018
Square roots of the intersection of A000290 and A268319. - Antti Karttunen, Jan 15 2019

			6^2 = (2^3 + 4^3)/2;
42^2 = (11^3 + 13^3)/2;
147^2 = (7^3 + 35^3)/2.

David A. Corneth, Table of n, a(n) for n = 1..3948
Zak Seidov, Triples {n,a,b} for n's up to 5*10^5

Cf. A000290, A000578, A268319.

Cf. also A273822.

nn = 13552; lim = Floor[(2 nn^2)^(1/3)]; Sort[Reap[Do[num = (a^3 + b^3)/2; If[IntegerQ[num] && num <= nn^2 && IntegerQ[Sqrt[num]], Sow[Sqrt[num]]], {a, lim}, {b, a - 1}]][[2, 1]]]
(* Second program: *)
Sqrt[#]&/@Select[Mean/@Subsets[Range[500]^3,{2}],IntegerQ[Sqrt[ #]]&]// Union (* Harvey P. Dale, Oct 13 2018 *)
upto[m_] := Module[{res = {}, n = m*m, i, j, k}, For[i = 1, i <= Floor[ Quotient[n, 2]^(1/3)], i++, For[j = i+2, j <= Floor[(n-i^3)^(1/3)], j += 2, If[IntegerQ[k = Sqrt[(i^3 + j^3)/2]], AppendTo[res, k]]]]; Sort[res]]; upto[20000] (* Jean-François Alcover, Jan 17 2019, after David A. Corneth *)

upto(n) = {my(res = List(), k); n*=n; for(i = 1, sqrtnint(n \ 2, 3), forstep(j = i + 2, sqrtnint(n - i^3, 3), 2, if(issquare((i^3 + j^3) / 2, &k),
listput(res, k)))); listsort(res); res} \\ David A. Corneth, Nov 25 2018

n^2 is average of two cubes: n^2 = (a^3 + b^3)/2, 0 < a < b.

Edited by M. F. Hasler, Dec 10 2018

A322102 Numbers that can be written as the average of two positive cubes in more than one way.

Original entry on oeis.org

2052, 6916, 16416, 19656, 32116, 32864, 55328, 55404, 82732, 85644, 131328, 157248, 160132, 186732, 256500, 256928, 262912, 402844, 442624, 443232, 477508, 497344, 504868, 508248, 530712, 537516, 546364, 597556, 661856, 665532, 685152, 703836, 783364, 804636

Offset: 1

Giovanni Resta, Nov 26 2018

nonn

The first term with 3 representations is 59912244, which is equal to (11^3 + 493^3)/2, (90^3 + 492^3)/2, and (346^3 + 428^3)/2.

			2052 is a term because it is equal to (2^3 + 16^3)/2 and (9^3 + 15^3)/2.

Cf. A268319, A267477.

Select[Range[10^5], Length[PowersRepresentations[2 #, 2, 3]] > 1 &]

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A186885 Numbers whose squares are the average of two distinct positive cubes.

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