A050802 -id:A050802 - cp's OEIS frontend

A050801 Numbers k such that k^2 is expressible as the sum of two positive cubes in at least one way.

3, 4, 24, 32, 81, 98, 108, 168, 192, 228, 256, 312, 375, 500, 525, 588, 648, 671, 784, 847, 864, 1014, 1029, 1183, 1225, 1261, 1323, 1344, 1372, 1536, 1824, 2048, 2187, 2496, 2646, 2888, 2916, 3000, 3993, 4000, 4200, 4225, 4536, 4563, 4644, 4704, 5184, 5324

Offset: 1

Patrick De Geest, Sep 15 1999

Analogous solutions exist for the sum of two identical cubes z^2 = 2*r^3 (e.g., 864^2 = 2*72^3). Values of 'z' are the terms in A033430, values of 'r' are the terms in A001105.
First term whose square can be expressed in two ways is 77976; 77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3. - Jud McCranie
First term whose square can be expressed in three ways is 3343221000; 3343221000^2 = 279300^3 + 2234400^3 = 790020^3 + 2202480^3 = 1256850^3 + 2094750^3.
First term whose square can be expressed in four ways <= 42794271007595289; 42794271007595289^2 = 14385864402^3 + 122279847417^3 = 55172161278^3 + 118485773289^3 = 64117642953^3 + 116169722214^3 = 96704977369^3 + 97504192058^3.
First term whose square can be expressed in five ways <= 47155572445935012696000; 47155572445935012696000^2 = 94405759361550^3 + 1305070263601650^3 = 374224408544280^3 + 1294899176535720^3 = 727959282778000^3 + 1224915311765600^3 = 857010857812200^3 + 1168192425418200^3 = 1009237516560000^3 + 1061381454915600^3.
After a(1) = 3 this is always composite, because factorization of the polynomial a^3 + b^3 into irreducible components over Z is a^3 + b^3 = (b+a)*(b^2 - ab + b^2). They may be semiprimes, as with 671 = 11 * 61, and 1261 = 13 * 97. The numbers can be powers in various ways, as with 32 = 2^5, 81 = 3^4, 256 = 2^8, 784 = 2^4 * 7^2 , 1225 = 5^2 * 7^2, and 2187 = 3^7. - Jonathan Vos Post, Feb 05 2011
If n is a term then n*b^3 is also a term for any b, e.g., 3 is a term hence 3*2^3 = 24, 3*3^3 = 81 and also 3*4^3 = 192 are terms. Sequence of primitive terms may be of interest. - Zak Seidov, Dec 11 2013
First noncubefree primitive term is 168 = 21*2^3 (21 is not a term of the sequence). - Zak Seidov, Dec 16 2013
From XU Pingya, Apr 10 2021: (Start)
Every triple (a, b, c) (with a^2 = b^3 + c^3) can produce a nontrivial parametric solution (x, y, z) of the Diophantine equation x^2 + y^3 + z^3 = d^4.
For example, to (1183, 65, 104), there is such a solution (d^2 - (26968032*d)*t^3 + 1183*8232^3*t^6, (376*d)*t - 65*8232^2*t^4, (92*d)*t - 104*8232^2*t^4).
To (77976, 228, 1824), there is (d^2 - (272916*d)*t^3 + 77976*57^3*t^6, (52*d)*t - 228*57^2*t^4, (74*d)*t - 1824*57^2*t^4).
Or to (77976, 1026, 1719), there is (d^2 - (25992*d)*t^3 + 77976*19^3*t^6, (37*d)*t - 1026*19^2*t^4, (11*d)*t - 1710*19^2*t^4). (End)

			1183^2 = 65^3 + 104^3.

Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..612 from T. D. Noe, terms 613..1000 from Harry J. Smith)

Cf. A050802, A000404, A033430, A001105, A038597, A050803, A106265, A217248.

A050801 := proc(n)
    option remember ;
    local a,x,y ;
    if n =1 then
        3
    else
        for a from procname(n-1)+1 do
            for x from 1 do
                if x^3 >= a^2 then
                    break ;
                end if;
                for y from 1 to x do
                    if x^3+y^3 = a^2 then
                        return a ;
                    end if;
                end do:
            end do:
        end do:
    end if;
end proc:
seq(A050801(n),n=1..20) ; # R. J. Mathar, Jan 22 2025

Select[Range[5350], Reduce[0 < x <= y && #^2 == x^3 + y^3, {x,y}, Integers] =!= False &] (* Jean-François Alcover, Mar 30 2011 *)
Sqrt[#]&/@Union[Select[Total/@(Tuples[Range[500],2]^3),IntegerQ[ Sqrt[ #]]&]] (* Harvey P. Dale, Mar 06 2012 *)
Select[Range@ 5400, Length@ DeleteCases[PowersRepresentations[#^2, 2, 3], w_ /; Times @@ w == 0] > 0 &] (* Michael De Vlieger, May 20 2017 *)

is(n)=my(N=n^2); for(k=sqrtnint(N\2,3),sqrtnint(N-1,3), if(ispower(N-k^3,3), return(n>1))); 0 \\ Charles R Greathouse IV, Dec 13 2013

a(n) = sqrt(A050802(n)). - Jonathan Sondow, Oct 28 2013

More terms from Michel ten Voorde and Jud McCranie

A050803 Cubes expressible as the sum of two nonzero squares in at least one way.

Original entry on oeis.org

8, 125, 512, 1000, 2197, 4913, 5832, 8000, 15625, 17576, 24389, 32768, 39304, 50653, 64000, 68921, 91125, 125000, 140608, 148877, 195112, 226981, 274625, 314432, 373248, 389017, 405224, 512000, 551368, 614125, 704969, 729000, 912673, 941192

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Root values equal terms from sequence A000404 'Sum of 2 nonzero squares'.
In other words, a(n)=(A000404(n))^3. - Artur Jasinski, Nov 29 2007
Obviously, if n and m are different members of this sequence, then n*m is also a member of this sequence. Additionally, if k^3 is a member of this sequence and k is not in A050804, then k^6 is also a member of this sequence. - Altug Alkan, May 11 2016

			551368 or 82^3 = 82^2 + 738^2 = 242^2 + 702^2.

Ian Stewart, "Game, Set and Math", Chapter 8 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

Index entries for sequences related to sums of squares

Cf. A000404, A050802, A050803, A135786, A135787, A135788.

a[n_]:=Module[{c=0},i=1; While[i^2Jayanta Basu, May 30 2013 *)
Select[Range[100]^3, Length[DeleteCases[PowersRepresentations[#, 2, 2], w_ /; MemberQ[w, 0]]] > 0 &] (* Michael De Vlieger, May 11 2016 *)

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

A217248 Numbers whose square is the sum of two nonnegative cubes.

Original entry on oeis.org

0, 1, 3, 4, 8, 24, 27, 32, 64, 81, 98, 108, 125, 168, 192, 216, 228, 256, 312, 343, 375, 500, 512, 525, 588, 648, 671, 729, 784, 847, 864, 1000, 1014, 1029, 1183, 1225, 1261, 1323, 1331, 1344, 1372, 1536, 1728, 1824, 2048, 2187, 2197, 2496, 2646, 2744, 2888

Offset: 1

Christian N. K. Anderson & Kevin L. Schwartz, Mar 20 2013

nonn

Numbers N such that N^2 = x^3 + y^3 where x and y are nonnegative integers. First case with 2 solutions is 77976^2 = 228^3 + 1824^3 = 1026^3 + 1710^3, see A051302. - Zak Seidov, Mar 21 2013

			312 is in the sequence because 312^2 = 2^3 + 46^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5000

This sequence with only positive (nonzero) cubes: A050801, and that sequence squared: A050802

A natural extension of the hypotenuse numbers A009003.

m = 2888; Sort[Reap[Do[If[IntegerQ[c = Sqrt[a^3 + b^3]], Sow[c]], {a, 0, m^(2/3)}, {b, a, (m^2 - a^3)^(1/3)}]][[2, 1]]] (* Zak Seidov, Mar 21 2013 *)

is(n)=n*=n;for(k=ceil((n/2-.5)^(1/3)),(n+.5)^(1/3),if(ispower(n-k^3,3),return(1)));0 \\ Charles R Greathouse IV, Mar 20 2013

y=c(); maxsol=3000 #All solutions x)==as.integer(x))y=c(y,x)
sort(y)

Offset and a(35) corrected and a(36)-a(51) from Giovanni Resta, Mar 20 2013

A230716 Numbers whose square is both a sum and a difference of two positive cubes.

Original entry on oeis.org

588, 1029, 1323, 2888, 4704, 8232, 8281, 9747, 10584, 15876, 23104, 27783, 33124, 35113, 35721, 37632, 47089, 65856, 66248, 73500, 74529, 77976, 84672, 103544, 114075, 127008, 127896, 128625, 165375, 184832, 201684, 222264, 223587, 263169, 264992, 280904

Offset: 1

Jonathan Sondow, Oct 28 2013

nonn

Intersection of A050801 and A038597.

a(5)-a(24) are computed from Donovan Johnson's extension of A230717.

			588^2 = 14^3 + 70^3 = 71^3 - 23^3.

Ian Stewart, "Game, Set and Math", Dover, 2007, Chapter 8 'Close Encounters of the Fermat Kind', pp. 107-124.

Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..500 n=1..100 from Donovan Johnson
Index to sequences related to sums of cubes

Cf. A050801, A050802, A038596, A038597, A230717.

a(n)^2 = a^3 + b^3 = c^3 - d^3 for some natural numbers a, b, c, d.

a(n) = sqrt(A230717(n)).

A230717 Squares that are both a sum and a difference of two positive cubes.

Original entry on oeis.org

345744, 1058841, 1750329, 8340544, 22127616, 67765824, 68574961, 95004009, 112021056, 252047376, 533794816, 771895089, 1097199376, 1232922769, 1275989841, 1416167424, 2217373921, 4337012736, 4388797504, 5402250000, 5554571841, 6080256576, 7169347584, 10721359936

Offset: 1

Jonathan Sondow, Oct 28 2013

nonn

Intersection of A050802 and A038596.

Square terms of sequence A225908. - Michel Marcus, Apr 22 2016

			345744 = 588^2 = 14^3 + 70^3 = 71^3 - 23^3.

Ian Stewart, "Game, Set and Math", Dover, 2007, Chapter 8 'Close Encounters of the Fermat Kind', pp. 107-124.

Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..500 n = 1..100 from Donovan Johnson
Index to sequences related to sums of cubes

Cf. A050801, A050802, A038596, A038597, A225908, A230716.

isA038596(n)=for(k=sqrtnint(n,3)+1,(sqrtint(12*n-3)+3)\6,if(ispower(n-k^3,3), return(issquare(n)))); 0
isA050802(n)=for(k=sqrtnint((n+1)\2, 3), sqrtnint(n-1, 3), if(ispower(n-k^3, 3), return(issquare(n)))); 0
is(n)=isA038596(n) && isA050802(n) \\ Charles R Greathouse IV, Oct 28 2013

a(n) = k^2 = a^3 + b^3 = c^3 - d^3 for some natural numbers k, a, b, c, d.

a(n) = A230716(n)^2.

a(5)-a(24) from Donovan Johnson, Oct 28 2013

A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

Original entry on oeis.org

9, 16, 128, 243, 576, 1024, 6561, 8192, 9604, 11664, 28224, 36864, 51984, 65536, 97344, 140625, 177147, 250000, 275625, 345744, 419904, 450241, 524288, 614656, 717409, 746496, 1028196, 1058841, 1399489, 1500625, 1590121, 1750329, 1806336, 1882384, 2359296, 3326976, 4194304

Offset: 1

Altug Alkan, Jan 10 2016

nonn

Intersection of A001597 and A003325.
Motivation for this sequence is the equation m^k = x^3 + y^3 where x,y,m > 0 and k >= 2.
Obviously, because of Fermat's Last Theorem, a(n) cannot be a cube.
A050802 is a subsequence.
Obviously, this sequence contains all numbers of the form 2^(3*n+1) and 3^(3*n-1), for n > 0.

			9 is a term because 9 = 3^2 = 1^3 + 2^3.
16 is a term because 16 = 2^4 = 2^3 + 2^3.
243 is a term because 243 = 3^5 = 3^3 + 6^3.

Chai Wah Wu, Table of n, a(n) for n = 1..5012

Cf. A001597, A003325, A050802, A085332, A013730, A013733, A266927.

T = thueinit('z^3+1);
is(n) = #select(v->min(v[1], v[2])>0, thue(T, n))>0;
for(n=2, 1e7, if(ispower(n) && is(n), print1(n, ", ")))

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A050801 Numbers k such that k^2 is expressible as the sum of two positive cubes in at least one way.

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A050803 Cubes expressible as the sum of two nonzero squares in at least one way.

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A217248 Numbers whose square is the sum of two nonnegative cubes.

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A230716 Numbers whose square is both a sum and a difference of two positive cubes.

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A230717 Squares that are both a sum and a difference of two positive cubes.

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A267088 Perfect powers of the form x^3 + y^3 where x and y are positive integers.

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