A217248 -id:A217248 - 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

A050802 Squares expressible as the sum of two positive cubes in at least one way.

Original entry on oeis.org

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

Offset: 1

Patrick De Geest, Sep 15 1999

			E.g., 717409 = 847^2 = 33^3 + 88^3.
169 = 13^2 = (-7)^3 + 8^3 is not a member, because 169 is not the sum of two positive cubes. - _Jonathan Sondow_, Oct 28 2013

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

Tony D. Noe and Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sums of squares

Cf. A038597, A050801, A050803, A106265, A217248.

ok[n_] := Length[Select[PowersRepresentations[n, 2, 3], #[[1]] != 0 & ]] >= 1; Select[Range[1600]^2, ok]
(* Jean-François Alcover, Apr 22 2011 *)
Union[Select[Total/@Tuples[Range[250]^3,2],IntegerQ[Sqrt[#]]&]] (* Harvey P. Dale, Mar 04 2012 *)

{ nstart=1; a2start=9; n=nstart; a=sqrtint(a2start)-1; until (0, a=a+1; a2=a*a; b1=((a2/2)^(1/3))\1; for (b=b1, a, b3=b*b*b; c1=1; if (a2 > b3, c1=((a2-b3)^(1/3))\1;); for (c=c1, b, d=b3 + c*c*c; if (d > a2 && c == 1, break(2)); if (d > a2, break); if (a2 == d, print(n, " ", a2); write("b050802.txt", n, " ", a2); n=n+1; break(2); ); ) ) ) } \\ Harry J. Smith, Jan 15 2009

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

a(n) = A050801(n)^2. - Jonathan Sondow, Oct 28 2013

More terms from Michel ten Voorde

Definition corrected by Jonathan Sondow, Oct 28 2013

A251781 Numbers whose square is the sum of two distinct positive cubes.

Original entry on oeis.org

3, 24, 81, 98, 168, 192, 228, 312, 375, 525, 588, 648, 671, 784, 847, 1014, 1029, 1183, 1225, 1261, 1323, 1344, 1536, 1824, 2187, 2496, 2646, 2888, 3000, 3993, 4200, 4225, 4536, 4563, 4644, 4704, 5184, 5368, 6156, 6272, 6292, 6371, 6591, 6696, 6776, 6877, 8112

Offset: 1

Daniel Arribas, Dec 08 2014

nonn

This list contains A117642 (if n=3*k^3, then n^2 = 9*k^6 = 8*k^6 + k^6 = (2*k^2)^3 + (k^2)^3). (Old comment rewritten as suggested by Michel Marcus, Dec 10 2014.)

Subsequence of A050801 and A217248. - Wolfdieter Lang, Jan 04 2015

			3^2 = 1^3 + 2^3; 24^2 = 4^3 + 8^3.

Daniel Arribas, Table of n, a(n) for n = 1..575

Cf. A024670, A117642, A050801, A217248, A099426 (coprime positive cubes).

def aupto(limit):
  c = [i**3 for i in range(1, int(limit**(2/3))+2) if i**3 <= limit**2]
  cc = [c1 + c2 for i, c1 in enumerate(c) for c2 in c[i+1:]]
  return sorted([i for i in range(1, limit+1) if i*i in cc])
print(aupto(8122)) # Michael S. Branicky, Mar 24 2021

L = []
for k in range(1,10^3):
    for l in range(k + 1,10^3):
        if is_square(k**3+l**3):
            L.append(sqrt(k**3+l**3))

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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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A050802 Squares expressible as the sum of two positive cubes in at least one way.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A251781 Numbers whose square is the sum of two distinct positive cubes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Sage