A003325 - OEIS

%I #103 Sep 20 2024 10:25:13

%S 2,9,16,28,35,54,65,72,91,126,128,133,152,189,217,224,243,250,280,341,

%T 344,351,370,407,432,468,513,520,539,559,576,637,686,728,730,737,756,

%U 793,854,855,945,1001,1008,1024,1027,1064,1072,1125,1216,1241,1332,1339,1343

%N Numbers that are the sum of 2 positive cubes.

%C It is conjectured that this sequence and A052276 have infinitely many numbers in common, although only one example (128) is known. [Any further examples are greater than 5 million. - _Charles R Greathouse IV_, Apr 12 2020] [Any further example is greater than 10^12. - _M. F. Hasler_, Jan 10 2021]

%C 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

%C From _James R. Buddenhagen_, Oct 16 2008: (Start)

%C (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,....

%C (ii) For 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.

%C 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. (End)

%C 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 the form n*m^3 with n = some previous term of the sequence and m >= 2) are 2, 9, 28, 35, 65, 91, 126, etc. - _Zak Seidov_, Oct 12 2011

%C This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - _Ant King_, May 09 2013

%C By Fermat's Last Theorem (the special case for exponent 3, proved by Euler, is sufficient), this sequence contains no cubes. - _Charles R Greathouse IV_, Apr 03 2021

%D C. G. J. Jacobi, Gesammelte Werke, vol. 6, 1969, Chelsea, NY, p. 354.

%H N. J. A. Sloane, <a href="/A003325/b003325.txt">Table of n, a(n) for n = 1..20000</a> (first 1000 terms from T. D. Noe)

%H F. Beukers, <a href="http://dx.doi.org/10.1215/S0012-7094-98-09105-0">The Diophantine equation Ax^p+By^q=Cz^r</a>, Duke Math. J. 91 (1998), 61-88.

%H Kevin A. Broughan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.html">Characterizing the sum of two cubes</a>, J. Integer Seqs., Vol. 6, 2003.

%H Nils Bruin, <a href="http://dx.doi.org/10.1007/10722028_9">On powers as sums of two cubes</a>, in Algorithmic number theory (Leiden, 2000), 169-184, Lecture Notes in Comput. Sci., 1838, Springer, Berlin, 2000.

%H C. G. J. Jacobi, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABR8803">Gesammelte Werke</a>.

%H Michael Penn, <a href="https://www.youtube.com/watch?v=RK1DewTkKT8">1674 is not a perfect cube</a>, 2020 video

%H N. J. A. Sloane, <a href="/A003325/a003325.txt">Table of n, a(n) for n = 1..59562</a>

%H D. Tournes, <a href="http://www.reunion.iufm.fr/dep/mathematiques/Seminaires/ActesPDF/Tournes53.pdf">A Glance on Indian Mathematician Srinivasa Ramanujan(1887-1920). [Text in French]</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>

%H <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>

%t 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 *)

%t With[{upto=2000},Select[Total/@Tuples[Range[Ceiling[Surd[upto,3]]]^3,2],#<=upto&]]//Union (* _Harvey P. Dale_, Jun 11 2016 *)

%o (PARI) cubes=sum(n=1, 11, x^(n^3), O(x^1400)); v = select(x->x, Vec(cubes^2), 1); vector(#v, k, v[k]+1) \\ edited by _Michel Marcus_, May 08 2017

%o (PARI) isA003325(n) = for(k=1,sqrtnint(n\2,3), ispower(n-k^3,3) && return(1)) \\ _M. F. Hasler_, Oct 17 2008, improved upon suggestion of _Altug Alkan_ and _Michel Marcus_, Feb 16 2016

%o (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

%o (PARI) list(lim)=my(v=List()); lim\=1; for(x=1,sqrtnint(lim-1,3), my(x3=x^3); for(y=1,min(sqrtnint(lim-x3,3),x), listput(v, x3+y^3))); Set(v) \\ _Charles R Greathouse IV_, Jan 11 2022

%o (Haskell)

%o a003325 n = a003325_list !! (n-1)

%o a003325_list = filter c2 [1..] where

%o    c2 x = any (== 1) $ map (a010057 . fromInteger) $

%o                        takeWhile (> 0) $ map (x -) $ tail a000578_list

%o -- _Reinhard Zumkeller_, Mar 24 2012

%o (Python)

%o from sympy import integer_nthroot

%o def aupto(lim):

%o   cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]

%o   sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])

%o   return [s for s in sum_cubes if s <= lim]

%o print(aupto(1343)) # _Michael S. Branicky_, Feb 09 2021

%Y Subsequence of A004999 and hence of A045980; supersequence of A202679.

%Y Cf. A024670 (2 distinct cubes), A003072, A001235, A011541, A003826, A010057, A000578, A027750, A010052, A085323 (n such that a(n+1)=a(n)+1).

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_

%E Error in formula line corrected by _Zak Seidov_, Jul 23 2009