

A050801


Numbers n such that n^2 is expressible as the sum of two positive cubes in at least one way.


12



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

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


REFERENCES

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107124.


LINKS

T. D. Noe and Harry J. Smith, Table of n, a(n) for n = 1..1000


FORMULA

a(n) = sqrt(A050802(n)).  Jonathan Sondow, Oct 28 2013


EXAMPLE

1183^2 = 65^3 + 104^3.


MATHEMATICA

Select[Range[5350], Reduce[0 < x <= y && #^2 == x^3 + y^3, {x, y}, Integers] =!= False &] (* JeanFrançois Alcover, Mar 30 2011 *)
Sqrt[#]&/@Union[Select[Total/@(Tuples[Range[500], 2]^3), IntegerQ[ Sqrt[ #]]&]] (* Harvey P. Dale, Mar 06 2012 *)


PROG

(PARI) nstart=1; astart=3; n=nstart; a=astart1; 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=((a2b3)^(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(N1, 3), if(ispower(Nk^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 (seqfan(AT)tenvoorde.org) and Jud McCranie


STATUS

approved



