A202679 -id:A202679 - cp's OEIS frontend

A003325 Numbers that are the sum of 2 positive cubes.

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

Offset: 1

N. J. A. Sloane

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]
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
From James R. Buddenhagen, Oct 16 2008: (Start)
(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 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. (End)
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
This is an infinite sequence in which the first term is prime but thereafter all terms are composite. - Ant King, May 09 2013
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

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

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
F. Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
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.
Michael Penn, 1674 is not a perfect cube, 2020 video
N. J. A. Sloane, Table of n, a(n) for n = 1..59562
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

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

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

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

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

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

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

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

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

from sympy import integer_nthroot
def aupto(lim):
  cubes = [i*i*i for i in range(1, integer_nthroot(lim-1, 3)[0] + 1)]
  sum_cubes = sorted([a+b for i, a in enumerate(cubes) for b in cubes[i:]])
  return [s for s in sum_cubes if s <= lim]
print(aupto(1343)) # Michael S. Branicky, Feb 09 2021

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

A020898 Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.

Original entry on oeis.org

2, 6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 49, 50, 51, 53, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 97, 98, 103, 105, 106, 107, 110, 114, 115, 117, 123, 124, 126, 127, 130

Offset: 1

Steven Finch

These numbers are the cubefree sums of two nonzero rational cubes.
This sequence does not contain A202679, which has members that are not cubefree. - Robert Israel, Mar 16 2016
Notice that 34^3 + 74^3 = 48*21^3 = 6*42^3 because 48 = 6*2^3 is not cubefree, but now 17^3 + 37^3 = 6*21^3 and 6 is already listed in the sequence. - Michael Somos, Mar 13 2023

			37^3 + 17^3 = 6*21^3 is the smallest positive solution for n = 6 (found by Lagrange).
5^3 + 4^3 = 7*3^3 is the smallest positive solution for n = 7.

B. N. Delone and D. K. Faddeev, The Theory of Irrationalities of the Third Degree, Amer. Math. Soc., 1964.
L. E. Dickson, History of The Theory of Numbers, Vol. 2, Chap. XXI, Chelsea NY 1966.
L. J. Mordell, Diophantine Equations, Academic Press, Chap. 15.

David W. Wilson, Table of n, a(n) for n = 1..255 (from Finch paper)
J. H. E. Cohn, The £ 450 question, Math. Mag., 73 (no. 3, 2000), 220-226.
Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Cf. A159843, A166246, A228499, A254324, A254326.

(* A naive program with a few pre-computed terms *) nmax = 130; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Reap[ Do[ n = CubeFreePart[ x*y*(x+y) ]; If[ 1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union; A020898 = Union[nn, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107, 123}](* Jean-François Alcover, Mar 30 2012 *)

Entry revised by N. J. A. Sloane, Aug 12 2004

Links updated by Max Alekseyev, Oct 17 2007 and Dec 12 2007

A228556 Sums of two coprime positive cubes that are also sums of two coprime positive fifth powers.

Original entry on oeis.org

2, 32769, 14348908, 14381675, 1073741825, 1088090731, 30517578126, 30517610893, 30531927032, 31591319949, 43977108474, 470184984577, 500702562701, 4747561509944, 4747561542711, 4747575858850, 4748635251767, 4778079088068, 5217746494519

Offset: 1

Arkadiusz Wesolowski, Aug 25 2013

nonn

Every term greater than 2 has at least one prime factor of the form 30*k + 1 and therefore is in A228541.

			14381675 is in the sequence since 32^3 + 243^3 = 8^5 + 27^5 = 14381675 and (32, 243) = (8, 27) = 1.

Arkadiusz Wesolowski and Chai Wah Wu, Table of n, a(n) for n = 1..103 (terms n = 1..24 from Arkadiusz Wesolowski)
Index to sequences related to sums of cubes

Cf. A035046, A202679, A228541, A228542.

A202679 INTERSECT A228542.

A295976 Number of nonnegative solutions to (x,y) = 1 and x^3 + y^3 = n.

Original entry on oeis.org

0, 2, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Seiichi Manyama, Dec 01 2017

nonn

Number of ordered pairs of two nonnegative natural numbers that are coprime and whose cubes add to n. - Antti Karttunen, May 31 2021

			For 1729, a(1729) = 4, because the following four ordered pairs, (1,12),  (9,10),  (10,9) and (12,1) satisfy the condition, as 1^3 + 12^3 = 9^3 + 10^3 = 1729. - _Antti Karttunen_, May 31 2021

Antti Karttunen, Table of n, a(n) for n = 0..65537

Cf. A173677, A225530, A295819, A295977.

Cf. also A202679.

{a(n) = sum(i=0, n, sum(j=0, n, if((gcd(i, j)==1) && (i^3+j^3==n), 1, 0)))}

A295976(n) = { my(s=0); for(i=0, oo, i3 = i^3; forstep(j=n-i3, 0, -1, if((i3+j^3==n) && gcd(i, j)==1, s++)); if(i3>n, return(s))); }; \\ Antti Karttunen, May 31 2021

A212286 Least k > 0 such that nk = x^3 + y^3 for nonnegative coprime x and y.

Original entry on oeis.org

1, 3, 7, 7, 21, 4, 19, 1, 37, 31, 39, 5, 2, 57, 67, 73, 7, 7, 103, 6, 133, 133, 147, 157, 18, 7, 1, 211, 237, 7, 259, 273, 301, 1, 13, 10, 4, 9, 403, 421, 3, 8, 487, 19, 541, 553, 579, 11, 637, 651, 9, 703, 31, 757, 26, 9, 853, 871, 903, 13, 27, 2, 1027, 1, 1101, 11, 1159, 1191

Offset: 2

Charles R Greathouse IV, Jun 12 2012

nonn

Broughan calls this eta(n) and proves that it exists for all n.

			4 is not the sum of two nonnegative cubes. 8 = 0^3 + 2^3, but 0 and 2 are not coprime. The least multiple of 4 that can be so represented is 28 = 1^3 + 3^3, so a(4) = 28/4.

Charles R Greathouse IV, Table of n, a(n) for n = 2..1000
Kevin A. Broughan, Characterizing the Sum of Two Cubes, J. Integer Seqs., Vol. 6, 2003.
Index entries for sequences related to sums of cubes

Cf. A004999, A212285, A202679.

sumOfTwoCubes(n)=my(k1=ceil((n-1/2)^(1/3)), k2=floor((4*n+1/2)^(1/3)), L); fordiv(n,d,if(d>=k1 && d<=k2 && denominator(L=(d^2-n/d)/3)==1 && issquare(d^2-4*L), return(1))); 0
sumOfTwoRPCubes(n)=if(sumOfTwoCubes(n),if(vecmax(factor(n)[,2])<3,1,for(x=ceil((n\2)^(1/3)),(n+.5)^(1/3),if(gcd(n,x)==1&&ispower(n-x^3,3),return(1)));0),0)
a(n)=forstep(k=n,2*n*(n^2+3),n,if(sumOfTwoRPCubes(k),return(k/n)))

a(n) <= 2n^2 + 6, a(a(n)) <= n.

A334964 Numbers that are the sum of three coprime positive cubes.

Original entry on oeis.org

3, 10, 17, 29, 36, 43, 55, 62, 66, 73, 92, 99, 118, 127, 129, 134, 141, 153, 155, 160, 179, 190, 197, 216, 218, 225, 244, 251, 253, 258, 277, 281, 307, 314, 342, 345, 349, 352, 359, 368, 371, 378, 397, 405, 408, 415, 433, 434, 466, 469, 471, 476, 495, 514, 521, 532, 540, 547, 557, 560, 566, 567

Offset: 1

Robert Israel, May 17 2020

nonn

The greatest common divisor of the three cubes must be 1, but they need not be pairwise coprime.

			a(3)=17 is in the sequence because 17 = 1^3 + 2^3 + 2^3 with gcd(1,2,2)=1.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of cubes

Cf. A202679.

N:= 1000: # for all terms <= N
S:= {seq(seq(seq(x^3+y^3+z^3, z=select(t -> igcd(x,y,t)=1, [$y..floor((N-x^3-y^3)^(1/3))])), y=x..floor(((N-x^3)/2)^(1/3))), x=1..floor((N/3)^(1/3)))}:
sort(convert(S, list));

list(lim)=my(v=List(),s,g,x3); lim\=1; if(lim<3, return([])); for(x=1,sqrtnint(lim\3,3), x3=x^3; for(y=x,sqrtnint((lim-x3)\2,3), s=x3+y^3; g=gcd(x,y); if(g>1, for(z=y,sqrtnint(lim-s,3), if(gcd(g,z)==1, listput(v,s+z^3))), for(z=y,sqrtnint(lim-s,3), listput(v,s+z^3))))); Set(v) \\ Charles R Greathouse IV, May 18 2020

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A003325 Numbers that are the sum of 2 positive cubes.

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A020898 Positive cubefree integers n such that the Diophantine equation X^3 + Y^3 = n*Z^3 has solutions.

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A228556 Sums of two coprime positive cubes that are also sums of two coprime positive fifth powers.

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A295976 Number of nonnegative solutions to (x,y) = 1 and x^3 + y^3 = n.

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A212286 Least k > 0 such that nk = x^3 + y^3 for nonnegative coprime x and y.

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A334964 Numbers that are the sum of three coprime positive cubes.

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