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A003325
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Numbers that are the sum of 2 positive cubes.
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43
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2, 9, 16, 28, 35, 54, 65, 72, 91, 126, 128, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 351, 370, 407, 432, 468, 513, 520, 539, 559, 576, 637, 686, 728, 730, 737, 756, 793, 854, 855, 945, 1001, 1008, 1024, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1343
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OFFSET
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1,1
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COMMENTS
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It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known.
A113958 is a subsequence; if m is a term then m+k^3 is a term of A003072 for all k>0. - Reinhard Zumkeller, Jun 03 2006
If n is a term then n*m^3 (m>=2) is also a term, e.g., 2m^3, 9m^3, 28m^3, and 35m^3 are all terms of the sequence. "Primitive" terms (not of form n*m^3 with n = some previous term of the sequence and m>=2) are 2,9,28,35,65,91,126 etc. - Moshe Levin, Oct 12 2011
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REFERENCES
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F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Nils Bruin, On powers as sums of two cubes, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.
C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..1000
C. G. J. Jacobi, Gesammelte Werke.
D. Tournes, A Glance on Indian Mathematician Srinivasa Ramanujan(1887-1920). [Text in French]
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes
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FORMULA
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Comment from James Buddenhagen, Oct 16 2008: (i) N and N+1 are both the sum of two positive cubes if N=2*(2*n^2+4*n+1)*(4*n^4+16*n^3+23*n^2+14*n+4), n=1,2,.... (ii) For integer n >= 2, let N = 16*n^6-12*n^4+6*n^2-2, so N+1 = 16*n^6-12*n^4+6*n^2-1. Then the identities 16*n^6-12*n^4+6*n^2-2 = (2*n^2-n-1)^3 + (2*n^2+n-1)^3 16*n^6-12*n^4+6*n^2-1 = (2*n^2)^3 + (2*n^2-1)^3 show that N, N+1 are in the sequence.
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MATHEMATICA
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nn = 2*20^3; Union[Flatten[Table[x^3 + y^3, {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]] (* T. D. Noe, Oct 12 2011 *)
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PROG
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(PARI) cubes=sum(n=1, 11, x^(n^3), O(x^1400)); print(cubes^2)
(PARI) isA003325(n) = for( k=1, sqrtn(n\2, 3), round(sqrtn(n-k^3, 3))^3+k^3==n & return(1)) [From M. F. Hasler, Oct 17 2008]
(Haskell)
a003325 n = a003325_list !! (n-1)
a003325_list = filter c2 [1..] where
c2 x = any (== 1) $ map (a010057 . fromInteger) $
takeWhile (> 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Mar 24 2012
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CROSSREFS
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Cf. A003072, A001235, A011541, A003826.
Cf. A085323 (n such that a(n+1)=a(n)+1). [From M. F. Hasler, Oct 17 2008]
Cf. A010057, A000578.
Sequence in context: A011193 A085960 A051386 * A101420 A213389 A178440
Adjacent sequences: A003322 A003323 A003324 * A003326 A003327 A003328
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Error in formula line corrected by Zak Seidov, Jul 23 2009
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STATUS
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approved
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