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A003325 Numbers that are the sum of 2 positive cubes. 50
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King May 09 2013

REFERENCES

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.

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

FORMULA

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.

MATHEMATICA

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 *)

PROG

(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)) \\ M. F. Hasler, Oct 17 2008

(PARI) T=thueinit('z^3+1);

is(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0 \\ Charles R Greathouse IV, Nov 29 2014

(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

CROSSREFS

Subsequence of A045980.

Cf. A003072, A001235, A011541, A003826, A010057, A000578, A027750, A010052, A004999, A085323 (n such that a(n+1)=a(n)+1).

Sequence in context: A011193 A085960 A051386 * A101420 A248434 A213389

Adjacent sequences:  A003322 A003323 A003324 * A003326 A003327 A003328

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Error in formula line corrected by Zak Seidov, Jul 23 2009

STATUS

approved

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Last modified December 19 16:42 EST 2014. Contains 252236 sequences.