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 A186885 Numbers whose squares are the average of two distinct positive cubes. 3
 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 (list; graph; refs; listen; history; text; internal format)
 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^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 (* 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

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Last modified October 15 13:06 EDT 2019. Contains 328030 sequences. (Running on oeis4.)