A020898 -id:A020898 - cp's OEIS frontend

A254324 Least Y such that X^3 + Y^3 = A020898(n)*Z^3 for some X <= Y and some Z.

1, 37, 5, 2, 89, 7, 683

Offset: 1

M. F. Hasler, Jan 27 2015

Also, max {X,Y,Z} for the smallest (in this sense of this sup norm) positive integer solution (X,Y,Z) to X^3 + Y^3 = A020898(n)*Z^3.
The X values are given in A254326.
a(8) > 10^5, with A020898(8)=17. Then the sequence continues a(9,10,...) = 5, 19, ?, 75, 3, 163, ?, 1853, ?, 3, 19, ...

			A020898(1)=2 and 1^3 + 1^3 = 2*1^3, therefore a(1)=1.
A020898(2)=6 and 17^3 + 37^3 = 6*21^3, and there is no "smaller" solution (with X, Y, Z < 37), therefore a(2)=37.

Cf. A020898, A254326.

a(n,L=10^9)={n=if(n>0,A020898[n],-n); for(b=1,L,for(a=1,b,(a^3+b^3)%n&&next;ispower((a^3+b^3)/n,3)&&return(b)))}

A254326 X value for the "least" solution (X,Y,Z) to X^3 + Y^3 = A020898(n)*Z^3, X <= Y.

Original entry on oeis.org

1, 17, 4, 1, 19, 2, 397

Offset: 1

M. F. Hasler, Jan 28 2015

The Y values, sup norms of the solution vector (X,Y,Z), are given in A254324.

			A020898(1)=2: 1^3 + 1^3 = 2*1^3,
A020898(2)=6: 17^3 + 37^3 = 6*21^3,
A020898(3)=7: 4^3 + 5^3 = 7*3^3,
A020898(4)=9: 1^3 + 2^3 = 9*1^3,
A020898(5)=12: 19^3 + 89^3 = 12*39^3,
A020898(6)=13: 2^3 + 7^3 = 13*3^3,
A020898(7)=15: 397^3 + 683^3 = 15*294^3, ...
For n >=9 the smallest solutions are: 3^3 + 5^3 = 19*2^3, 1^3 + 19^3 = 20*7^3, ?, 53^3 + 75^3 = 26*28^3, 1^3 + 3^3 = 28*1^3, 107^3 + 163^3 = 30*57^3, ?, 523^3 + 1853^3 = 33*582^3, ?, 2^3 + 3^3 = 35*1^3, 18^3 + 19^3 = 37*7^3, ?, 1^3 + 7^3 = 43*2^3, ...

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

A202679 Numbers that are sums of two coprime positive cubes.

Original entry on oeis.org

2, 9, 28, 35, 65, 91, 126, 133, 152, 189, 217, 341, 344, 351, 370, 407, 468, 513, 539, 559, 637, 730, 737, 793, 854, 855, 1001, 1027, 1072, 1241, 1332, 1339, 1343, 1358, 1395, 1456, 1547, 1674, 1729, 1843, 1853, 2060, 2071, 2198, 2205, 2224, 2261, 2322, 2331, 2413

Offset: 1

Arkadiusz Wesolowski, Jan 06 2012

nonn

Not a subsequence of A020898: non-cubefree members of this sequence include 152, 189, 344, 351, 513, 1072. - Robert Israel, Mar 16 2016

			28 is in the sequence since 1^3 + 3^3 = 28 and (1, 3) = 1.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
R. C. Baker, Sums of two relatively prime cubes, Acta Arithmetica 129(2007), 103-146.
Kevin A. Broughan, A computational approach to characterizing the sum of two cubes, Hamilton: University of Waikato, 2001, p. 9.
P. Erdős and K. Mahler, On the number of integers which can be represented by a binary form, J. London Math. Soc. 13 (1938), pp. 134-139. [alternate link]
P. Erdős, On the integers of the form x^k + y^k, J. London Math. Soc. 14 (1939), pp. 250-254.
Index to sequences related to sums of cubes

Subsequence of A003325.

Cf. A024670, A001235, A018850.

N:= 10000: # to get all terms <= N
S:= {2,seq(seq(x^3 + y^3, y = select(t -> igcd(t,x)=1, [$x+1 .. floor((N - x^3)^(1/3))])), x = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S,list)); # Robert Israel, Mar 15 2016

nn = 2500; Union[Flatten[Table[If[CoprimeQ[x, y] == True, x^3 + y^3, {}], {x, nn^(1/3)}, {y, x, (nn - x^3)^(1/3)}]]]
Select[Range@ 2500, Length[PowersRepresentations[#, 2, 3] /. {{0, } -> Nothing, {a, b_} /; ! CoprimeQ[a, b] -> Nothing}] > 0 &] (* Michael De Vlieger, Mar 15 2016 *)

is(n)=for(k=1,(n\2+.5)^(1/3),if(gcd(k,n)==1&&ispower(n-k^3, 3), return(1)));0 \\ Charles R Greathouse IV, Apr 13 2012

list(lim)=my(v=List()); forstep(x=1, lim^(1/3), 2, forstep(y=2,(lim-x^3+.5)^(1/3), 2, if(gcd(x,y)==1, listput(v,x^3+y^3))); forstep(y=1, min((lim-x^3+.5)^(1/3),x), 2, if(gcd(x,y)==1, listput(v,x^3+y^3)))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Dec 05 2012

Erdős & Mahler shows that a(n) < kn^(3/2) for some k. Erdős later gives an elementary proof. - Charles R Greathouse IV, Dec 05 2012

A166246 Primes representable as the sum of two rational cubes.

Original entry on oeis.org

2, 7, 13, 17, 19, 31, 37, 43, 53, 61, 67, 71, 79, 89, 97, 103, 107, 127, 139, 151, 157, 163, 179, 193, 197, 211, 223, 229, 233, 241, 251, 269, 271, 277, 283, 313, 331, 337, 349, 359, 367, 373, 379, 397, 409, 421, 431, 433, 439, 449, 457, 463, 467, 499, 503, 521

Offset: 1

Max Alekseyev, Oct 10 2009

nonn

The prime elements of A159843, i.e., the intersection of A159843 and A000040.

Also, the prime elements of A020898.

H. Cohen, Number Theory. I, Tools and Diophantine Equations, Springer-Verlag, 2007, p. 378.

Fernando Rodriguez Villegas, Don Zagier, Which primes are sums of two cubes?, CMS Conference Proceedings 15 (1995), pp. 295-306.

Cf. A166243, A166244, A159843.

(* To speed up computation, a few terms are pre-computed *) nmax = 521; xmax = 360; preComputed = {127, 271, 379}; solQ[p_] := Do[ If[ IntegerQ[z = Root[-x^3 - y^3 + p*#^3 & , 1]], Print[p, {x, y, z}]; Return[True]], {x, 2, xmax}, {y, x, xmax}]; A166246 = Union[ preComputed, Select[ Prime[ Range[ PrimePi[nmax]]], Mod[#, 9] == 4 || Mod[#, 9] == 7 || Mod[#, 9] == 8 || solQ[#] === True & ]](* Jean-François Alcover, Apr 04 2012, after given formula *)

Under the Birch and Swinnerton-Dyer conjecture, these primes consist of:
(i) p = 2;
(ii) p == 4, 7, or 8 (mod 9);
(iii) p == 1 (mod 9) and p divides A206309(p-1), i.e., Villegas-Zagier polynomial A166243((p-1)/3) evaluated at x=0.

A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

Original entry on oeis.org

1, 37, 2, 2, 89, 7, 683, 18, 3, 19, 25469, 3, 3, 163, 137, 1853, 631, 3, 4, 449, 7, 11, 23417, 730511, 1872, 28747, 5, 11, 4, 4, 5353, 2538163, 15409, 53, 197, 17351, 5563, 13, 433, 2570129, 13, 1176498611, 53, 1241, 4, 25903, 15642626656646177, 14, 5, 592, 4033, 165889, 90, 181, 9109, 5266097, 5, 184223499139, 5, 5, 7, 52954777

Offset: 1

Jean-François Alcover, May 11 2011

nonn

This sequence a(k) is computed so that equation a(k)^3 + y^3 = A020898(k)*z^3 holds.
The 4 sequences A020898 [i.e., n], A190356 [i.e., x], A190580 [i.e., y] and A190581 [i.e., z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3.
All x values above 25469 were obtained from Mishima's list and may not be the least positive solution.

			a(18) = 3 because A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190580, A190581

(* Let x = u + v and y = u - v *)
f[n_, m_] := (r =  Reduce[u > 0 && v > 0 && Mod[2*u^3 + 6*u*v^2, n] == 0, {u, v},  Integers] ;
uv={u,v}/.(ToRules/@ List@@ r[[All,-2;;-1]])/.C-> c;
xy = (s = {};
Do[sel =  Select[uv,  IntegerQ[((2*#1[[1]]^3 + 6*#1[[1]]*#1[[2]]^2)/n)^(1/ 3)] &];
If[sel =!= {}, AppendTo[s, sel] ], {c[1], 0, m}, {c[2], 0,  m}];
{#[[1]] + #[[2]], #[[1]] - #[[2]]} & /@ (s //
Flatten[#, 1] &)) // Select[#, Total[#] != 0 &] &;
nxyz =  xy /. {x_Integer, y_} -> {n, x, y, ((x^3 + y^3)/n)^(1/3)};
nxyz /. ({, x, y_, z_} /; {x, y, z} != {0, 0, 0} &&
GCD[x, y, z] != 1) :> (gd = GCD[x, y, z]; {n, x/gd, y/gd, z/gd})) // Union // Sort[#, #1[[2]] < #2[[2]] &] &;
g[n_] := (m0 = 1; While[(r = f[n, m0]) == {}, m0 = 2 m0];
r // First);
A020898 = {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}; km = Length[A020898]; (* xm(n) = some hard to compute values of x from Hisanori Mishima's list *) xm[22]=25469; xm[50]=23417; xm[51]=730511; xm[58]=28747; xm[68]=2538163; xm[69]=15409; xm[75]=17351; xm[85]=2570129; xm[87]=1176498611; xm[92]=25903; xm[94]=15642626656646177; xm[106]=165889; xm[114]=9109; xm[115]=5266097; xm[123]=184223499139; xm[130]=52954777; xm[n_] := xm[n] = g[n][[2]];
A190356 = Table[ n = A020898[[k]]; Print[xm[n]]; xm[n], {k, 1, km}] (* Jean-François Alcover, Jan 03 2012 *)

Positions corresponding to n=124 and n=127 (which were not minimal) corrected by Jean-François Alcover

Extended to 62 terms by Jean-François Alcover, Jan 03 2012

A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

Original entry on oeis.org

1, 17, -1, 1, 19, 2, 397, -1, -2, 1, 17299, -1, 1, 107, -65, 523, -359, 2, -3, -71, 1, -2, -11267, 62641, -1819, -14653, -4, 7, -1, 1, 1208, -472663, -10441, 17, -126, -11951, 53, -4, 323, -2404889, 5, -907929611, 36, -431, 3, -3547, -15616184186396177, -5, -3, -349, 3527, -140131, 17, -71, -901, -2741617, -2, 10183412861, -1, 1, -6, 33728183

Offset: 1

Jean-François Alcover, May 13 2011

sign

A190356(n)^3 + a(n)^3 = A020898(n)*z^3. Unknown z corresponds to sequence A190581.

The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356^3 + A190580^3 = A020898 * A190581^3

			a(18) = 2  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190356, A190581.

Table[ y /. First[ Solve[ A190356[[n]]^3 + y^3 == A020898[[n]] * A190581[[n]]^3 ] ], {n, 62}] (* Jean-François Alcover, Jan 04 2012 *)

A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

Original entry on oeis.org

1, 21, 1, 1, 39, 3, 294, 7, 1, 7, 9954, 1, 1, 57, 42, 582, 182, 1, 1, 129, 2, 3, 6111, 197028, 217, 7083, 1, 3, 1, 1, 1323, 620505, 3318, 13, 43, 3606, 1302, 3, 111, 330498, 3, 216266610, 13, 273, 1, 5733, 590736058375050, 3, 1, 117, 1014, 25767, 19, 37, 1878, 1029364, 1, 37045412880, 1, 1, 1, 11285694

Offset: 1

Jean-François Alcover, May 13 2011

nonn

A190356(n)^3 + y^3 = A020898(n)*a(n)^3. Unknown y corresponds to sequence A190580.

The 4 sequences A020898 [i.e. n], A190356 [i.e. x], A190580 [i.e. y] and A190581 [i.e. z] satisfy the equation A190356(n)^3 + A190580(n)^3 = A020898(n) * A190581(n)^3

			a(18) = 1  because  A020898(18) = 35 and 3^3 + 2^3 = 35*1^3.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Nakao Hisayasu, Rational Points on Elliptic Curves: x^3+y^3=n (nna2.html up to nna22.html)
Hisanori Mishima, Solutions of Diophantine equation x^3+y^3=A.z^3 ...

Cf. A020898, A060838, A190356, A190580

a[n_] := z /. ToRules[ Reduce[ z > 0 && A190356[[n]]^3 + A190580[[n]]^3 == A020898[[n]]*z^3, z, Integers]]; Table[a[n] , {n, 1, 62}]

A228485 Odd prime powers p^k such that p is congruent to 2 or 5 mod 9.

Original entry on oeis.org

5, 11, 23, 25, 29, 41, 47, 59, 83, 101, 113, 121, 125, 131, 137, 149, 167, 173, 191, 227, 239, 257, 263, 281, 293, 311, 317, 347, 353, 383, 389, 401, 419, 443, 461, 479, 491, 509, 529, 563, 569, 587, 599, 617, 625, 641, 653, 659, 677, 743, 761, 797, 821, 839

Offset: 1

Arkadiusz Wesolowski, Aug 23 2013

nonn

For any n, the equation x^3 + y^3 = a(n)*z^3 is not solvable in nonzero integers. Therefore, these numbers do not occur in A020898.

Henri Cohen, Number Theory. Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, Springer, 2007, pp. 374-375.

Wikipedia, Prime power

Cf. A020898, A025473. Subsequence of A061345.

forstep(n=3, 839, 2, p=isprimepower(n); if(p>0, m=Mod(round(n^(1/p)), 9); if(m==2||m==5, print1(n, ", "))));

A228499 Sums of two rational cubes, excluding cubes and twice cubes.

Original entry on oeis.org

6, 7, 9, 12, 13, 15, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 56, 58, 61, 62, 63, 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, 120, 123, 124

Offset: 1

Arkadiusz Wesolowski, Aug 23 2013

nonn

Each term can be written as sum of two rational cubes infinitely many times.

These are all the integers A>0 such that the rank of the elliptic curve x^3 + y^3 = A is positive (A060838(A)>0). - Michael Somos, Feb 29 2020

Wacław Sierpiński, Teoria liczb, cz. II, PWN, Warsaw, 1959, pp. 472-473.

Index entries for sequences related to sums of cubes

Subsequence of A020897, and hence of A159843.

Cf. A020898, A060838.

for(n=1, 124, if(ellanalyticrank(ellinit([0, (4*n)^2]))[1]>0, print1(n, ", ")));

cp's OEIS Frontend

A254324 Least Y such that X^3 + Y^3 = A020898(n)*Z^3 for some X <= Y and some Z.

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A254326 X value for the "least" solution (X,Y,Z) to X^3 + Y^3 = A020898(n)*Z^3, X <= Y.

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A159843 Sums of two rational cubes.

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A202679 Numbers that are sums of two coprime positive cubes.

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A166246 Primes representable as the sum of two rational cubes.

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A190356 Least positive x in the Diophantine equation x^3 + y^3 = n*z^3 (with x >= y and y != 0).

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A190580 Value of y in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

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A190581 Value of z in the Diophantine equation x^3 + y^3 = n*z^3 (with x>0 and minimal and x >= y and y != 0).

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