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 A051302 Numbers whose square can be expressed as the sum of two positive cubes in more than one way. 6
 77976, 223587, 623808, 894348, 1788696, 2105352, 2989441, 4298427, 4672423, 4990464, 5986575, 6036849, 7154784, 8437832, 9747000, 14309568, 16842816, 23915528, 24147396, 24770529, 26745768, 27948375, 34387416, 34634719, 36570744, 37379384, 39923712, 47892600 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Observations regarding terms through a(64)=306761364: All are multiples of 7^2, 13^2, and/or 19^2. Other than 2, 3, 5 and 11, their only prime factors are 7, 13, 19, 31, 43, 61, 67, 79, 127, 151, and 181 (each of which exceeds a multiple of 6 by 1). None is a cube or higher power; the ones that are squares are a(7), a(12), a(17), a(19), a(20), a(32), a(33), a(41), a(49), a(55), and a(58). - Jon E. Schoenfield, Oct 08 2006 Many of the terms beyond a(64) have prime factors other than those found in a(1) through a(64); however, each term through a(774) has at most one distinct prime factor p > 5 that does not exceed a multiple of 6 by 1, and p, if such a prime factor exists, has a multiplicity m=3, with only a few exceptions: n=651 and n=713 (where p^m is 11^2), n=346 and n=770 (where p^m is 17^2), n=699 and n=740 (where p^m is 23^2), and n=741 (where p^m is 11^6). - Jon E. Schoenfield, Oct 20 2013 First differs from A145553 at A051302(172)=3343221000 where 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3. This sequence is the union of A145553 and A155961. This sequence is infinite. If n is a member of this sequence, then n^2 = a^3 + b^3 = c^3 + d^3 where (a, b) and (c, d) are distinct pairs. If n^2 = a^3 + b^3 = c^3 + d^3, then (n*k^3)^2 = n^2*k^6 = k^6*(a^3 + b^3) = k^6*(c^3 + d^3) = (a*k^2)^3 + (b*k^2)^3 = (c*k^2)^3 + (d*k^2)^3. It is obvious that if (a, b) and (c, d) are distinct, then (k^2*a, k^2*b), (k^2*c, k^2*d) are also distinct for all nonzero values of k. So if n is in this sequence, then n*k^3 is in this sequence for all k > 0. - Altug Alkan, May 10 2016 LINKS Jon E. Schoenfield and Ray Chandler, Table of n, a(n) for n = 1..774 EXAMPLE 2989441^2 = 1729^3+20748^3 = 15561^3+17290^3, so 2989441 is in the sequence. MATHEMATICA (* Warning: this script is only a recomputation of the original b-file of 64 terms from Jon E. Schoenfield, and should not be used to extend the data. *) max = 310000000; cubeFreeParts = {361, 8281, 33124, 159201, 169309, 221725, 565068, 628849, 917427, 1054729, 2370963, 2989441, 4672423, 8968323, 9402967, 9795747, 34634719}; r[x_] := Reduce[0 < y <= z && x^2 == y^3 + z^3, {y, z}, Integers]; okQ[primes_] := Intersection[{2, 3, 5, 7, 11, 13, 19, 31, 43, 61, 67, 79, 127, 139, 151, 181}, primes] == primes; crop[n_] := Reap[For[m = 1, True, m++, x = n*m^3; If[x > max, Break[]]; If[okQ[FactorInteger[x][[All, 1]]], If[Head[rx = r[x]] === Or, Print["x = ", x, " ", rx]; Sow[x]]; ]]][[2, 1]]; A051302 = crop /@ cubeFreeParts // Flatten // Sort (* Jean-François Alcover, Jul 02 2017 *) PROG (PARI) T=thueinit('x^3+1, 1); is(n)=my(v=thue(T, n^2)); sum(i=1, #v, v[i][1]>=0 && v[i][2]>=v[i][1])>1 \\ Charles R Greathouse IV, May 10 2016 CROSSREFS Cf. A050801, A001235, A011541, A145553, A155961. Sequence in context: A266967 A061528 A210141 * A145553 A016819 A016855 Adjacent sequences:  A051299 A051300 A051301 * A051303 A051304 A051305 KEYWORD nonn,nice AUTHOR EXTENSIONS Definition corrected by Jon E. Schoenfield, Aug 27 2006 More terms from Jon E. Schoenfield, Oct 08 2006 Extended by Ray Chandler, Nov 22 2011 STATUS approved

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Last modified January 19 15:13 EST 2022. Contains 350466 sequences. (Running on oeis4.)