A273822 -id:A273822 - 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

A274578 Nonsquare k such that k^3 - 1 is the average of two positive cubes.

Original entry on oeis.org

2305, 2629, 4117, 7060, 37444, 46081, 113320, 208545, 449569, 474553, 507325, 1224757, 1499068, 1927405, 1931077, 2263129, 2350909, 2447596, 3107841, 4065517, 4274932, 4303321, 5646685, 6582865, 7225597, 10386273, 18432001, 21936709, 24218425, 24362989, 27351417

Offset: 1

Altug Alkan, Jun 29 2016

nonn

The equation x^3 + y^3 = 2*z^3 has no integer solution triple (x, y, z) for x > y and z is nonzero. So this sequence focuses on the equation x^3 + y^3 = 2*(z^3 - 1) where x, y > 0.

			2305 is a term because it is not a square and 2305^3 - 1 = (144^3 + 2904^3) / 2.

Chai Wah Wu, Table of n, a(n) for n = 1..66

Cf. A000037, A003325, A068601, A273822.

isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
lista(nn) = for(n=1, nn, if(isA003325(2*(n^3-1)) && !issquare(n), print1(n, ", ")));

a(9)-a(25) from Chai Wah Wu, Aug 07 2020

a(26)-a(31) from Chai Wah Wu, Jun 30 2025

cp's OEIS Frontend

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

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