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 A050801 Numbers k such that k^2 is expressible as the sum of two positive cubes in at least one way. 22
 3, 4, 24, 32, 81, 98, 108, 168, 192, 228, 256, 312, 375, 500, 525, 588, 648, 671, 784, 847, 864, 1014, 1029, 1183, 1225, 1261, 1323, 1344, 1372, 1536, 1824, 2048, 2187, 2496, 2646, 2888, 2916, 3000, 3993, 4000, 4200, 4225, 4536, 4563, 4644, 4704, 5184, 5324 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Analogous solutions exist for the sum of two identical cubes z^2 = 2*r^3 (e.g., 864^2 = 2*72^3). Values of 'z' are the terms in A033430, values of 'r' are the terms in A001105. First term whose square can be expressed in two ways is 77976; 77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3. - Jud McCranie First term whose square can be expressed in three ways is 3343221000; 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3. First term whose square can be expressed in four ways <= 42794271007595289; 42794271007595289^2 = 14385864402^3 + 122279847417^3 = 55172161278^3 + 118485773289^3 = 64117642953^3 + 116169722214^3 = 96704977369^3 + 97504192058^3. First term whose square can be expressed in five ways <= 47155572445935012696000; 47155572445935012696000^2 = 94405759361550^3 + 1305070263601650^3 = 374224408544280^3 + 1294899176535720^3 = 727959282778000^3 + 1224915311765600^3 = 857010857812200^3 + 1168192425418200^3 = 1009237516560000^3 + 1061381454915600^3. After a(1) = 3 this is always composite, because factorization of the polynomial a^3 + b^3 into irreducible components over Z is a^3 + b^3 = (b+a)*(b^2 - ab + b^2). They may be semiprimes, as with 671 = 11 * 61, and 1261 = 13 * 97. The numbers can be powers in various ways, as with 32 = 2^5, 81 = 3^4, 256 = 2^8, 784 = 2^4 * 7^2 , 1225 = 5^2 * 7^2, and 2187 = 3^7. - Jonathan Vos Post, Feb 05 2011 If n is a term then n*b^3 is also a term for any b, e.g., 3 is a term hence 3*2^3 = 24, 3*3^3 = 81 and also 3*4^3 = 192 are terms. Sequence of primitive terms may be of interest. - Zak Seidov, Dec 11 2013 First noncubefree primitive term is 168 = 21*2^3 (21 is not a term of the sequence). - Zak Seidov, Dec 16 2013 From XU Pingya, Apr 10 2021: (Start) Every triple (a, b, c) (with a^2 = b^3 + c^3) can produce a nontrivial parametric solution (x, y, z) of the Diophantine equation x^2 + y^3 + z^3 = d^4. For example, to (1183, 65, 104), there is such a solution (d^2 - (26968032*d)*t^3 + 1183*8232^3*t^6, (376*d)*t - 65*8232^2*t^4, (92*d)*t - 104*8232^2*t^4). To (77976, 228, 1824), there is (d^2 - (272916*d)*t^3 + 77976*57^3*t^6, (52*d)*t - 228*57^2*t^4, (74*d)*t - 1824*57^2*t^4). Or to (77976, 1026, 1719), there is (d^2 - (25992*d)*t^3 + 77976*19^3*t^6, (37*d)*t - 1026*19^2*t^4, (11*d)*t - 1710*19^2*t^4). (End) REFERENCES Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..612 from T. D. Noe, terms 613..1000 from Harry J. Smith) FORMULA a(n) = sqrt(A050802(n)). - Jonathan Sondow, Oct 28 2013 EXAMPLE 1183^2 = 65^3 + 104^3. MATHEMATICA Select[Range, Reduce[0 < x <= y && #^2 == x^3 + y^3, {x, y}, Integers] =!= False &] (* Jean-François Alcover, Mar 30 2011 *) Sqrt[#]&/@Union[Select[Total/@(Tuples[Range, 2]^3), IntegerQ[ Sqrt[ #]]&]] (* Harvey P. Dale, Mar 06 2012 *) Select[Range@ 5400, Length@ DeleteCases[PowersRepresentations[#^2, 2, 3], w_ /; Times @@ w == 0] > 0 &] (* Michael De Vlieger, May 20 2017 *) PROG (PARI) nstart=1; astart=3; n=nstart; a=astart-1; until (0, a++; a2=a^2; b1=((a2/2)^(1/3))\1; for (b=b1, a, b3=b^3; c1=1; if (a2 > b3, c1=((a2-b3)^(1/3))\1; ); for (c=c1, b, d=b3 + c^3; if (d > a2 && c == 1, break(2)); if (d > a2, break); if (a2 == d, print(n, " ", a); write("b050801.txt", n, " ", a); n++; break(2))))) \\ Harry J. Smith, Jan 15 2009 (PARI) is(n)=my(N=n^2); for(k=sqrtnint(N\2, 3), sqrtnint(N-1, 3), if(ispower(N-k^3, 3), return(n>1))); 0 \\ Charles R Greathouse IV, Dec 13 2013 CROSSREFS Cf. A050802, A000404, A033430, A001105, A038597, A050803, A106265, A217248. Sequence in context: A032831 A047180 A051394 * A103093 A124632 A048091 Adjacent sequences:  A050798 A050799 A050800 * A050802 A050803 A050804 KEYWORD nonn,nice AUTHOR Patrick De Geest, Sep 15 1999 EXTENSIONS More terms from Michel ten Voorde and Jud McCranie STATUS approved

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Last modified September 26 00:53 EDT 2022. Contains 356986 sequences. (Running on oeis4.)