A020897 -id:A020897 - cp's OEIS frontend

A003336 Numbers that are the sum of 2 positive 4th powers.

2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6562, 6577, 6642, 6817, 7186, 7857, 8192, 8962, 10001, 10016, 10081, 10256, 10625

Offset: 1

N. J. A. Sloane

Numbers k such that k = x^4 + y^4 has a solution in positive integers x, y.
There are no squares in this sequence. - Altug Alkan, Apr 08 2016
As the order of addition doesn't matter we can assume terms are in nondecreasing order. - David A. Corneth, Aug 01 2020

			From _David A. Corneth_, Aug 01 2020: (Start)
16378801 is in the sequence as 16378801 = 43^4 + 60^4.
39126977 is in the sequence as 39126977 = 49^4 + 76^4.
71769617 is in the sequence as 71769617 = 19^4 + 92^4. (End)

Sean A. Irvine, Table of n, a(n) for n = 1..20000 (terms 1..1000 from T. D. Noe, terms 1001..10000 from David A. Corneth)
A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
S. R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On Generalized Fermat-Wiles Equation [From the Wayback Machine]
Samuel S. Wagstaff, Jr., Equal Sums of Two Distinct Like Powers, J. Int. Seq., Vol. 25 (2022), Article 22.3.1.
Eric Weisstein's World of Mathematics, Biquadratic Number.

5906 is the first term in A060387 but not in this sequence. Cf. A020897.
Cf. A088687 (2 distinct 4th powers).
A###### (x, y): Numbers that are the form of x nonzero y-th powers.
Cf. A000404 (2, 2), A000408 (3, 2), A000414 (4, 2), A003072 (3, 3), A003325 (3, 2), A003327 (4, 3), A003328 (5, 3), A003329 (6, 3), A003330 (7, 3), A003331 (8, 3), A003332 (9, 3), A003333 (10, 3), A003334 (11, 3), A003335 (12, 3), A003336 (2, 4), A003337 (3, 4), A003338 (4, 4), A003339 (5, 4), A003340 (6, 4), A003341 (7, 4), A003342 (8, 4), A003343 (9, 4), A003344 (10, 4), A003345 (11, 4), A003346 (12, 4), A003347 (2, 5), A003348 (3, 5), A003349 (4, 5), A003350 (5, 5), A003351 (6, 5), A003352 (7, 5), A003353 (8, 5), A003354 (9, 5), A003355 (10, 5), A003356 (11, 5), A003357 (12, 5), A003358 (2, 6), A003359 (3, 6), A003360 (4, 6), A003361 (5, 6), A003362 (6, 6), A003363 (7, 6), A003364 (8, 6), A003365 (9, 6), A003366 (10, 6), A003367 (11, 6), A003368 (12, 6), A003369 (2, 7), A003370 (3, 7), A003371 (4, 7), A003372 (5, 7), A003373 (6, 7), A003374 (7, 7), A003375 (8, 7), A003376 (9, 7), A003377 (10, 7), A003378 (11, 7), A003379 (12, 7), A003380 (2, 8), A003381 (3, 8), A003382 (4, 8), A003383 (5, 8), A003384 (6, 8), A003385 (7, 8), A003387 (9, 8), A003388 (10, 8), A003389 (11, 8), A003390 (12, 8), A003391 (2, 9), A003392 (3, 9), A003393 (4, 9), A003394 (5, 9), A003395 (6, 9), A003396 (7, 9), A003397 (8, 9), A003398 (9, 9), A003399 (10, 9), A004800 (11, 9), A004801 (12, 9), A004802 (2, 10), A004803 (3, 10), A004804 (4, 10), A004805 (5, 10), A004806 (6, 10), A004807 (7, 10), A004808 (8, 10), A004809 (9, 10), A004810 (10, 10), A004811 (11, 10), A004812 (12, 10), A004813 (2, 11), A004814 (3, 11), A004815 (4, 11), A004816 (5, 11), A004817 (6, 11), A004818 (7, 11), A004819 (8, 11), A004820 (9, 11), A004821 (10, 11), A004822 (11, 11), A004823 (12, 11), A047700 (5, 2).
Cf. A000583 (4th powers).

nn=12; Select[Union[Plus@@@(Tuples[Range[nn],{2}]^4)], # <= nn^4&] (* Harvey P. Dale, Dec 29 2010 *)
Select[Range@ 11000, Length[PowersRepresentations[#, 2, 4] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 08 2016 *)

list(lim)=my(v=List()); for(x=1, sqrtnint(lim\=1,4), for(y=1, min(sqrtnint(lim-x^4,4), x), listput(v, x^4+y^4))); Set(v) \\ Charles R Greathouse IV, Apr 24 2012; updated July 13 2024

T=thueinit('x^4+1,1);
is(n)=#thue(T,n)>0 && !issquare(n) \\ Charles R Greathouse IV, Feb 26 2017

def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  return sorted(p2)
print(aupto(10625)) # Michael S. Branicky, Mar 18 2021

{i: A216284(i) > 0}. - R. J. Mathar, Jun 04 2021

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

A060387 Numbers k such that x^4 + y^4 = k * z^4 is solvable in nonzero integers x,y,z.

Original entry on oeis.org

2, 17, 32, 82, 97, 162, 257, 272, 337, 512, 626, 641, 706, 881, 1250, 1297, 1312, 1377, 1552, 1921, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 5906

Offset: 1

Michel ten Voorde, Apr 04 2001

Numbers k such that k = x^4 + y^4 has a solution in positive rationals x, y.

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
Steven R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Eric Weisstein's World of Mathematics, Biquadratic Number

5906 is the first term not in A003336. Cf. A020897.

Cf. A111152, A209431.

r[n_, z_] := Reduce[0 < x <= y && x^4 + y^4 == n*z^4, {x, y}, Integers]; zm[] = 1; zm[5906] = 17; ok[n] := (tf = False; Do[ If[ r[n, z] =!= False, tf = True; Break[]], {z, 1, zm[n]}]; tf); A060387 = Reap[ Do[ If[ ok[n], Print[n]; Sow[n]], {n, 1, 5906}]][[2, 1]](* Jean-François Alcover, Mar 09 2012 *)

A020896 Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.

Original entry on oeis.org

2, 31, 33, 64, 211, 242, 244, 275, 486, 781, 992, 1023, 1025, 1056, 1267, 2048, 2101, 2882, 3093, 3124, 3126, 3157, 3368, 4149, 4651, 6250, 6752, 7533, 7744, 7775, 7777, 7808, 8019, 8800, 9031, 10901, 13682, 15552, 15783, 15961, 16564

Offset: 0

Steven Finch

68101 = (15/2)^5 + (17/2)^5 is believed to be the smallest positive integer k which is the sum of two nonzero fifth powers of rational numbers but not the sum of two nonzero fifth powers of integers.

			31 = 2^5 + (-1)^5.

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
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. A001481, A020897, A003336.

Select[Union[Total/@(Select[Tuples[Range[-8,8],{2}], !MemberQ[#, 0]&]^5)],#>0&]  (* Harvey P. Dale, Apr 03 2011 *)

See Theorem 3.5.6 of J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.

A209431 Numbers n such that x^4 + y^4 = n * z^4 is solvable in nonzero integers x,y,z with z > 1 and gcd(x,y,n) = 1.

Original entry on oeis.org

5906, 469297, 926977, 952577, 1127857, 1298257, 1347361, 1647377, 2455361, 3342817, 4928977, 5268706, 5519537, 8588161, 8879537, 9339361, 9391537, 9846017, 11414017, 14543026, 15547297, 16502722, 16657217, 16672322, 16830017, 19730162, 23672002, 25030097, 27681937, 27979762

Offset: 1

Jean-François Alcover, Mar 09 2012

nonn

Values of z (1, 17, 41, 73, 89, ...) are elements of sequence A004625 (divisible only by primes congruent to 1 mod 8). The first composite z is 697 = 17*41: 41^4 + 822091^4 = 1935300738962*697^4.

Proof (after Ms. Adina Calvo) that values of z are divisible only by primes congruent to 1 mod 8: Let {x,y,z} be a nontrivial solution and p an odd prime divisor of z. Reducing the equation mod p, one gets in Z/pZ: x^4 + y^4 = 0 mod p. Hence (x*y^-1)^4 = -1, then x*y^-1 is an order-8 element of the multiplicative group (Z/pZ)*, which has p-1 elements. Therefore p is congruent to 1 mod 8.

			5906 is in the sequence because a^4 + b^4 = 5906*c^4 has the solution (a,b,c) = (25,149,17).

A. Bremner and P. Morton, A new characterization of the integer 5906, Manuscripta Math. 44 (1983) 187-229; Math. Rev. 84i:10016.
Steven R. Finch, On a generalized Fermat-Wiles equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Eric Weisstein's World of Mathematics, Biquadratic Number

Cf. A060387, A003336, A020897.

BiquadraticFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 4]} & /@ FactorInteger[n]); max = 10000; Sort[ Reap[Do[nz4 = x^4 + y^4; z4 = nz4/BiquadraticFreePart[nz4]; z = z4^(1/4); n = nz4/z4; If[z4 > 1 && IntegerQ[z] && GCD[x, y, z] == 1, Print[{n, x, y, z}]; Sow[n]], {x, 1, max}, {y, x, max}]][[2, 1]]]

Numbers in A060387 but not in A003336.

Definition corrected by Hugo Pfoertner, Nov 08 2016

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, ", ")));

A260823 Positive integers that are not divisible by any cube greater than 1 and cannot be written as the sum of two cubes of rational numbers.

Original entry on oeis.org

3, 4, 5, 10, 11, 14, 18, 21, 23, 25, 29, 36, 38, 39, 41, 44, 45, 46, 47, 52, 55, 57, 59, 60, 66, 73, 74, 76, 77, 82, 83, 93, 95, 99, 100, 101, 102, 109, 111, 113, 116, 118, 119, 121, 122, 129, 131, 137, 138, 145, 146, 147, 148, 149, 150, 154, 155, 158, 165

Offset: 1

Marco Ripà, Jul 31 2015

This sequence is infinite.

This sequence is the complement of (A020897 minus A046099), except 1.

			a(4)=10 cannot be written as c^3 + d^3 where both c and d are rational numbers.
22 = (25469/9954)^3 + (17299/9954)^3, so 22 is not in the sequence.

W. Sierpiński, 250 Problems in Elementary Number Theory, 1970, page 112.

Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203-362.

Cf. A003325, A020897, A046099.

cp's OEIS Frontend

A003336 Numbers that are the sum of 2 positive 4th powers.

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A159843 Sums of two rational cubes.

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A060387 Numbers k such that x^4 + y^4 = k * z^4 is solvable in nonzero integers x,y,z.

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A020896 Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.

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A209431 Numbers n such that x^4 + y^4 = n * z^4 is solvable in nonzero integers x,y,z with z > 1 and gcd(x,y,n) = 1.

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

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A260823 Positive integers that are not divisible by any cube greater than 1 and cannot be written as the sum of two cubes of rational numbers.

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