A045980 -id:A045980 - cp's OEIS frontend

A222304 Numbers k such that 2k is in A045980.

0, 1, 4, 8, 13, 14, 27, 28, 32, 36, 49, 62, 63, 64, 76, 104, 108, 109, 112, 125, 140, 148, 158, 171, 172, 185, 193, 216, 224, 234, 244, 252, 256, 260, 288, 301, 302, 343, 351, 364, 365, 378, 392, 427, 433, 468, 494, 496, 500, 504, 508, 512, 532, 536, 589, 603, 608, 652, 665, 666, 676, 679, 728, 729, 734, 756, 769, 832

Offset: 1

N. J. A. Sloane, Feb 15 2013, based on a suggestion from Allan C. Wechsler

nonn

Numbers k such that 2k = x^3 + y^3 for some integers x, y. - Charles R Greathouse IV, Nov 29 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A045980, A222305, A222306.

T=thueinit('z^3+1);
is(n)=n==0 || #thue(T, 2*n)>0 \\ Charles R Greathouse IV, Nov 29 2014

A222305 Numbers k such that 9k is in A045980.

Original entry on oeis.org

0, 1, 3, 6, 7, 8, 13, 14, 21, 24, 27, 31, 38, 39, 43, 48, 52, 56, 57, 64, 73, 78, 81, 84, 91, 95, 104, 105, 111, 112, 125, 133, 134, 147, 148, 155, 157, 162, 168, 183, 186, 189, 192, 195, 206, 211, 216, 237, 241, 244, 245, 248, 258, 259, 273, 291, 294, 301, 304, 305, 307, 312, 343, 344, 351, 372, 375, 378, 381, 384

Offset: 1

N. J. A. Sloane, Feb 15 2013, based on a suggestion from Allan C. Wechsler

nonn

The sequence "numbers k such that 3k is in A045980" consists of these numbers multiplied by 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A045980, A222304, A222306.

T=thueinit('z^3+1);is(n)=n==0 || #thue(T, 9*n)>0 \\ Charles R Greathouse IV, Nov 29 2014

A222306 Numbers k such that 4k is in A045980.

Original entry on oeis.org

0, 2, 4, 7, 14, 16, 18, 31, 32, 38, 52, 54, 56, 70, 74, 79, 86, 108, 112, 117, 122, 126, 128, 130, 144, 151, 182, 189, 196, 234, 247, 248, 250, 252, 254, 256, 266, 268, 304, 326, 333, 338, 364, 367, 378, 416, 430, 432, 434, 436, 448, 486, 500, 511, 515, 518, 542, 549, 556, 558, 560, 592, 632, 635, 662, 670, 679, 682

Offset: 1

N. J. A. Sloane, Feb 15 2013, based on a suggestion from Allan C. Wechsler

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A045980, A222304, A222305.

T=thueinit('z^3+1);is(n)=n==0 || #thue(T, 4*n)>0 \\ Charles R Greathouse IV, Nov 29 2014

A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

Original entry on oeis.org

9, 26, 35, 65, 91, 133, 169, 215, 217, 218, 335, 341, 386, 407, 469, 485, 511, 559, 721, 737, 793, 817, 866, 973, 1027, 1115, 1141, 1241, 1261, 1267, 1339, 1343, 1385, 1387, 1538, 1603, 1685, 1727, 1843, 1853, 1981, 2071, 2189, 2402, 2413, 2611, 2743, 2771

Offset: 1

Hugo Pfoertner, Jun 25 2003

nonn

			a(1)=9 because 2^3+1^3=3*3, a(2)=26=3^3-1^3=2*13.
a(5)=91 is the smallest semiprime expressible in two different ways: 91=4^3+3^3=6^3-5^3=7*13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A045980, A085366.

T=thueinit('z^3+1);
is(n)=bigomega(n)==2 && #thue(T, n)
list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), if(#thue(T, p*q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Nov 29 2014

A003325 Numbers that are the sum of 2 positive cubes.

Original entry on oeis.org

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

A004999 Sums of two nonnegative cubes.

Original entry on oeis.org

0, 1, 2, 8, 9, 16, 27, 28, 35, 54, 64, 65, 72, 91, 125, 126, 128, 133, 152, 189, 216, 217, 224, 243, 250, 280, 341, 343, 344, 351, 370, 407, 432, 468, 512, 513, 520, 539, 559, 576, 637, 686, 728, 729, 730, 737, 756, 793, 854, 855, 945, 1000, 1001

Offset: 1

N. J. A. Sloane, Steven Finch

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.
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
Index entries for sequences related to sums of cubes

Subsequence of A045980; A003325 is a subsequence.
Cf. A000578, A004825, A010057, A373972 (characteristic function).
Indices of nonzero terms in A025446.

a004999 n = a004999_list !! (n-1)
a004999_list = filter c2 [1..] where
   c2 x = any (== 1) $ map (a010057 . fromInteger) $
                       takeWhile (>= 0) $ map (x -) $ tail a000578_list
-- Reinhard Zumkeller, Dec 20 2013

Union[(#[[1]]^3+#[[2]]^3)&/@Tuples[Range[0,20],{2}]] (* Harvey P. Dale, Dec 04 2010 *)

is(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
list(lim)=my(v=List());for(x=0,(lim+.5)^(1/3),for(y=0,min(x,(lim-x^3)^(1/3)),listput(v,x^3+y^3))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jun 12 2012

is(n)=my(L=sqrtnint(n-1,3)+1,U=sqrtnint(4*n,3));fordiv(n,m,if(L<=m&m<=U,my(ell=(m^2-n/m)/3);if(denominator(ell)==1&&issquare(m^2-4*ell),return(1))));0 \\ Charles R Greathouse IV, Apr 16 2013

T=thueinit('z^3+1);
is(n)=n==0 || #select(v->min(v[1],v[2])>=0, thue(T,n))>0 \\ Charles R Greathouse IV, Nov 29 2014

A159843 Sums of two rational cubes.

Original entry on oeis.org

1, 2, 6, 7, 8, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 27, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117

Offset: 1

Steven Finch, Apr 23 2009

Conjectured asymptotic (based on the random matrix theory) is given in Cohen (2007) on p. 378.
The prime elements are listed in A166246. - Max Alekseyev, Oct 10 2009
Alpöge et al. prove 'that the density of integers expressible as the sum of two rational cubes is strictly positive and strictly less than 1.' The authors remark that it is natural to conjecture that these integers 'have natural density exactly 1/2.' - Peter Luschny, Nov 30 2022
Jha, Majumdar, & Sury prove that every nonzero residue class mod p (for prime p) has infinitely many elements, as do 1 and 8 mod 9. - Charles R Greathouse IV, Jan 24 2023
Alpöge, Bhargava, & Shnidman prove that the lower density of this sequence is at least 2/21 and its upper density is at most 5/6. - Charles R Greathouse IV, Feb 15 2023

H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 379.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Levent Alpöge, Manjul Bhargava, and Ari Shnidman, Integers expressible as the sum of two rational cubes, arXiv:2210.10730 [math.NT], Oct. 2022.
Bogdan Grechuk, Existence of Non-Trivial Solutions to Homogeneous Equations, Polynomial Diophantine Equations, Springer, Cham (2024), Chapter 6, 473-536.
Somnath Jha, Dipramit Majumdar, and B. Sury, Infinitely many primes in each of the residue classes 1 and 8 modulo 9 are sums of two rational cubes, arXiv preprint arXiv:2301.06970 [math.NT], 2023-2024.
Index entries for sequences related to sums of cubes

Complement of A185345.
Subsequences include A045980, A004999, and A003325.
Cf. A020894, A020895, A020897, A020898.

(* A naive program with a few pre-computed terms *) nmax = 117; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Join[{1}, Reap[ Do[n = CubeFreePart[x*y*(x + y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union]; A159843 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* Jean-François Alcover, Apr 03 2012 *)

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E=ellinit([0, 16*r^2]), eri=ellrankinit(E), mwr=ellrank(eri), ar); if(r<3 || mwr[1], return(1)); if(mwr[2]<1, return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(ar)); for(effort=1,99, mwr=ellrank(eri,effort); if(mwr[1]>0, return(1), mwr[2]<1, return(0))); "yes under BSD conjecture" \\ Charles R Greathouse IV, Dec 02 2022

A cubefree integer c>2 is in this sequence iff the elliptic curve y^2=x^3+16*c^2 has positive rank. - Max Alekseyev, Oct 10 2009

A152043 Numbers expressible as the difference of two nonnegative cubes.

Original entry on oeis.org

0, 1, 7, 8, 19, 26, 27, 37, 56, 61, 63, 64, 91, 98, 117, 124, 125, 127, 152, 169, 189, 208, 215, 216, 217, 218, 271, 279, 296, 316, 331, 335, 342, 343, 386, 387, 397, 448, 469, 485, 488, 504, 511, 512, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 729, 784

Offset: 1

Mark Taggart (mt2612f(AT)aol.com), Nov 21 2008

nonn

Subsequence of A045980. - R. J. Mathar, Nov 28 2008

Contains A000578 as a subsequence. - Chandler

			E.g. 7=2^3-1^3, 8=2^3-0^3, 296=8^3-6^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A042965, A152044, A152045.

The Index to the OEIS lists many related sequences under "difference of two cubes". - N. J. A. Sloane, Dec 04 2008

T=thueinit('z^3+1);
is(n)=n==0 || #select(v->v[1]<=0&&v[2]>=0, thue(T, n))>0 \\ Charles R Greathouse IV, Nov 29 2014

Extended by Ray Chandler, Dec 04 2008

A020894 Nonnegative numbers that are sums of two nonzero cubes.

Original entry on oeis.org

0, 2, 7, 9, 16, 19, 26, 28, 35, 37, 54, 56, 61, 63, 65, 72, 91, 98, 117, 124, 126, 127, 128, 133, 152, 169, 189, 208, 215, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 432, 448, 468, 469

Offset: 1

Steven Finch

nonn

From Michael B. Porter, Oct 16 2009: (Start)
When calculating terms, there is no need to search beyond a value x defined by x^3 - (x-1)^3 = n. The positive solution is given by x = 1/2 + (sqrt(12n-3))/6.
There are no cubes in this sequence, but the numbers before and after a cube are all included. (End)

			From _Michael B. Porter_, Oct 16 2009: (Start)
7 is in the sequence because 2^3 + (-1)^3 = 7
8 is not in the sequence because the only solutions to x^3 + y^3 = 8 have either x=0 or y=0. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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. A045980 [From Michael B. Porter, Oct 16 2009]

Reap[For[n = 0, n < 500, n++, fi = FindInstance[x > 0 && y != 0 && n == x^3 + y^3, {x, y}, Integers, 1]; If[fi =!= {}, Print[n, " = ", Hold[x^3 + y^3] /. fi[[1]]]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 05 2016 *)

isA020894(n) = {r=0;x=1.0/2+sqrt(12*n-3.0)/6;for(i=1,floor(x),if(ispower(n-i^3,3) & (n != i^3),r++));r>0}; \\ Michael B. Porter, Oct 16 2009

T=thueinit('z^3+1);
is(n)=n==0 || #select(v->v[1] && v[2], thue(T, n))>0 \\ Charles R Greathouse IV, Nov 29 2014

Definition and offset edited by N. J. A. Sloane, Dec 01 2009

A307585 Positive sums of two distinct cubes (of arbitrary sign).

Original entry on oeis.org

1, 7, 8, 9, 19, 26, 27, 28, 35, 37, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 448, 468, 469, 485, 488, 504, 511, 512, 513, 520, 539, 547, 559

Offset: 1

Robert Israel, Apr 15 2019

nonn

All terms == 0, 1, 2, 7 or 8 (mod 9).

			a(3) = 8 = 0^3 + 2^3.
a(4) = 9 = 1^3 + 2^3.
a(5) = 19 = (-2)^3 + 3^3.

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

Contained in A045980.  Contains A024670.
Primes in this sequence: A002407.
Cf. A060464.

filter:= proc(n) local d, dp, r;
   for d in numtheory:-divisors(n) do
     dp:= n/d;
     r:= 12*dp - 3*d^2;
     if r > 0 and issqr(r) and (sqrt(r)/6 + d/2)::integer then return true fi
   od;
   false
end proc:
select(filter, [$0..1000]);

filterQ[n_] := Module[{d, dp, r}, Catch[Do[dp = n/d; r = 12 dp - 3 d^2; If[r > 0 && IntegerQ[Sqrt[r]] && IntegerQ[Sqrt[r]/6 + d/2], Throw[True]], {d, Divisors[n]}]; False]];
Select[Range[1000], filterQ] (* Jean-François Alcover, Oct 17 2020, after Maple *)

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A222304 Numbers k such that 2k is in A045980.

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A222305 Numbers k such that 9k is in A045980.

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A222306 Numbers k such that 4k is in A045980.

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A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

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A003325 Numbers that are the sum of 2 positive cubes.

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A004999 Sums of two nonnegative cubes.

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A159843 Sums of two rational cubes.

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A152043 Numbers expressible as the difference of two nonnegative cubes.

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