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A050801
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Numbers n such that n^2 is expressible as the sum of two positive cubes in at least one way.
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7
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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
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OFFSET
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1,1
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COMMENTS
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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 that can be expressed in two ways: 77976^2 = 228^3+1824^3 = 1026^3+1710^3 - Jud McCranie.
First term that can be expressed in three ways: 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
First term that can be expressed in four ways <= 42794271007595289 where 42794271007595289^2 = 14385864402^3 + 122279847417^3 = 55172161278^3 + 118485773289^3 = 64117642953^3 + 116169722214^3 = 96704977369^3 + 97504192058^3.
First term that can be expressed in five ways <= 47155572445935012696000 where 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 5, 2011].
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REFERENCES
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Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.
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LINKS
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T. D. Noe and Harry J. Smith, Table of n, a(n) for n=1,...,1000
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EXAMPLE
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E.g. 1183^2 = 65^3 + 104^3.
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MATHEMATICA
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Select[Range[5350], Reduce[0 < x <= y && #^2 == x^3 + y^3, {x, y}, Integers] =!= False &] (* From Jean-François Alcover, Mar 30 2011 *)
Sqrt[#]&/@Union[Select[Total/@(Tuples[Range[500], 2]^3), IntegerQ[ Sqrt[ #]]&]] (* From Harvey P. Dale, Mar 06 2012 *)
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PROG
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(PARI) { nstart=1; astart=3; n=nstart; a=astart-1; until (0, a=a+1; a2=a*a; b1=((a2/2)^(1/3))\1; for (b=b1, a, b3=b*b*b; c1=1; if (a2 > b3, c1=((a2-b3)^(1/3))\1; ); for (c=c1, b, d=b3 + c*c*c; if (d > a2 && c == 1, break(2)); if (d > a2, break); if (a2 == d, print(n, " ", a); write("b050801.txt", n, " ", a); n=n+1; break(2); ); ) ) ) } [From Harry J. Smith, Jan 15 2009]
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CROSSREFS
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Cf. A050802, A000404, A033430, A001105.
Sequence in context: A032831 A047180 A051394 * A103093 A124632 A048091
Adjacent sequences: A050798 A050799 A050800 * A050802 A050803 A050804
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KEYWORD
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nonn,nice
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AUTHOR
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Patrick De Geest, Sep 15 1999.
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EXTENSIONS
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More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) and Jud McCranie.
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STATUS
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approved
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