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

1,1

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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 *)

Prog

(PARI) 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

Crossrefs

Cf. A000290, A000578, A268319.

Cf. also A273822.

Sequence in context: A306429 A117693 A153243 * A097253 A083938 A292316

Adjacent sequences: A186882 A186883 A186884 * A186886 A186887 A186888

Keyword

nonn

Author

Zak Seidov, Feb 28 2011

Extensions

Edited by M. F. Hasler, Dec 10 2018

Status

approved