A038597 -id:A038597 - cp's OEIS frontend

A038593 Differences between positive cubes in 1, 2 or 3 ways: union of A014439, A014440 and A014441.

7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 784, 817, 819, 866, 875, 919, 936, 973, 988

Offset: 1

Jeff Burch

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A038594, A038595, A038596, A038597, A038598.

N:= 1000: # to get all terms <= N
X:= floor(sqrt(N/3)):
V:= Vector(N):
for x from 2 to X do
  if x^3 > N then
     y0:= iroot(x^3-N,3);
     if x^3 - y0^3 > N then y0:= y0+1 fi;
  else y0:= 1 fi;
  for y from y0 to x-1 do
     V[x^3 - y^3] := V[x^3 - y^3]+1
  od
od:
select(t -> V[t] <= 3 and V[t]>=1, [$1..N]); # Robert Israel, Dec 10 2015

r = 988; p = 3; Sort@Drop[Flatten@Select[Tally@Reap[Do[n = i^p - j^p; If[n <= r, Sow[n]], {i, Ceiling[(r/p)^(1/(p - 1))]}, {j, i}]][[2, 1]], 0 < #[[2]] < 4 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Corrected by Don Reble, Nov 19 2006

A181123 Numbers that are the differences of two positive cubes.

Original entry on oeis.org

0, 7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 784, 817, 819, 866, 875, 919, 936, 973

Offset: 1

T. D. Noe, Oct 06 2010

Because x^3-y^3 = (x-y)(x^2+xy+y^2), the difference of two cubes is a prime number only if x=y+1, in which case all the primes are cuban, see A002407.
The difference can be a square (see A038597), but Fermat's Last Theorem prevents the difference from ever being a cube. Beal's Conjecture implies that there are no higher odd powers in this sequence.
If n is in the sequence, it must be x^3-y^3 where 0 < y <= x < n^(1/2). - Robert Israel, Dec 24 2017

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 from Noe)

Cf. A014439, A014440, A014441, A333376, A333377, A038593 (subsequence), A086120.

Cf. A024352 (squares), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

N:= 10^4: # to get all terms <= N
sort(convert(select(`<=`, {0, seq(seq(x^3-y^3, y=1..x-1),x=1..floor(sqrt(N)))}, N),list)); # Robert Israel, Dec 24 2017

nn=10^5; p=3; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
With[{nn=60},Take[Union[Abs[Flatten[Differences/@Tuples[ Range[ nn]^3,2]]]], nn]] (* Harvey P. Dale, May 11 2014 *)

list(lim)=my(v=List([0]),a3); for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim,sqrtnint(a3-lim-1,3)+1,1), a-1, listput(v,a3-b^3))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

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

Original entry on oeis.org

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

A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 15, 18, 19, 20, 23, 25, 26, 27, 28, 35, 39, 40, 44, 45, 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 64, 67, 71, 72, 74, 76, 79, 81, 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 125, 126, 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153

Offset: 1

Zak Seidov, Apr 28 2005

nonn

A given a(n) can have multiple solutions with distinct (b,c), e.g., a=4 with b=2, c=2 (4 + 2^2 = 2^3) or with b=11, c=5 (4 + 11^2 = 5^3). (See also A181138.) Sequences A106266 and A106267 list the minimal values. - M. F. Hasler, Oct 04 2013
The cubes A000578 = (1, 8, 27, 64, ...) form a subsequence of this sequence, corresponding to b=0, a=c^3. If b=0 is excluded, these terms are not present, except for a few exceptions, a = 216, 343, 12167, ... (6^3 + 28^2 = 10^3, 7^3 + 13^2 = 8^3, 23^3 + 588^2 = 71^3, ...), cf. A038597 for the possible b-values. - M. F. Hasler, Oct 05 2013
This is the complement of A081121. The values do indeed correspond to solutions listed in Gebel's file. - M. F. Hasler, Oct 05 2013
B-file corrected following a remark by Alois P. Heinz, May 24 2019. A double-check would be appreciated in view of two values that were missing, for unknown reasons, in the earlier version of the b-file. - M. F. Hasler, Aug 10 2024

			a = 1,2,4,7,8,11,13,15,18,19,20,23,25,26,27,28,35,39,40,44,45,47,48,49,53, ...
b = 0,5,2,1,0, 4,70, 7, 3,18,14, 2,10, 1, 0, 6,36, 5,52, 9,96,13,4,524,26, ...
c = 1,3,2,2,2, 3,17, 4, 3, 7, 6, 3, 5, 3, 3, 4,11, 4,14, 5,21, 6, 4,65, 9, ...
Here are the values grouped together:
{{1, 0, 1}, {2, 5, 3}, {4, 2, 2}, {7, 1, 2}, {8, 0, 2}, {11, 4, 3}, {13, 70, 17}, {15, 7, 4}, {18, 3, 3}, {19, 18, 7}, {20, 14, 6}, {23, 2, 3}, {25, 10, 5}, {26, 1, 3}, {27, 0, 3}, {28, 6, 4}, {35, 36, 11}, {39, 5, 4}, {40, 52, 14}, {44, 9, 5}, {45, 96, 21}, {47, 13, 6}, {48, 4, 4}, {49, 524, 65}, {53, 26, 9}, {54, 17, 7}, {55, 3, 4}, {56, 76, 18}, {60, 2, 4}, {61, 8, 5}, {63, 1, 4}, {64, 0, 4}, {67, 110, 23}, {71, 21, 8}, ... }
a(2243) = 10000 = 25^3 - 75^2. - _M. F. Hasler_, Oct 05 2013, index corrected Aug 10 2024
a(136) = 366 = 11815^3 - 1284253^2 (has c/a(n) ~ 32.3); a(939) = 3607 = 244772^3 - 121099571^2 (has c/a(n) ~ 67.9); a(1090) = 4265 = 84521^3 - 24572364^2 (has c/a(n) ~ 19.8). - _M. F. Hasler_, Aug 10 2024

M. F. Hasler, Table of n, a(n) for n = 1..2500 (corrected and extended Aug 10 2024)
J. Gebel, Integer points on Mordell curves, negative k values [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A106266, A106267 for respective minimal values of b and c.
Cf. A023055: (Apparent) differences between adjacent perfect powers (integers of form a^b, a >= 1, b >= 2); A076438: n which appear to have a unique representation as the difference of two perfect powers; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1; A076440: n which appear to have a unique representation as the difference of two perfect powers and one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = n, with a>0, b>0, x>1, y>1 and that solution has odd x or odd y (or both odd); A075772: Difference between n-th perfect power and the closest perfect power, etc.
Cf. A023055, A075772, A076438, A076440.
Cf. A054504, A081121, A081120; A179386 - A179388.

f[n_] := Block[{k = Floor[n^(1/3) + 1]}, While[k < 10^6 && !IntegerQ[ Sqrt[k^3 - n]], k++ ]; If[k == 10^6, 0, k]]; Select[ Range[ 154], f[ # ] != 0 &] (* Robert G. Wilson v, Apr 28 2005 *)

select( {is_A106265(a, L=99)=for(c=sqrtnint(a, 3), (a+9)*L, issquare(c^3-a, &b) && return(c))}, [1..199]) \\ The function is_A106265 returns 0 if n isn't a term, or else the c-value (A106267) which can't be zero if n is a term. The L-value can be used to increase the search limit but so far no instance is known that requires L>68. - M. F. Hasler, Aug 10 2024

a(n) = A106267(n)^3 - A106266(n)^2.

More terms from Robert G. Wilson v, Apr 28 2005

Definition corrected, solutions with b=0 added by M. F. Hasler, Sep 30 2013

A038596 Squares that are a difference between 2 positive cubes.

Original entry on oeis.org

169, 784, 2401, 10816, 21609, 32761, 35721, 50176, 123201, 130321, 150544, 153664, 257049, 301401, 345744, 456976, 571536, 692224, 1058841, 1382976, 1750329, 1874161, 2030625, 2096704, 2286144, 2640625, 3211264, 3467044, 3651921, 3694084, 5285401, 5546025

Offset: 1

Jeff Burch

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A038593-A038598.

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

mm=820188; cb=vector(mm); for(i=1, mm, cb[i]=i^3); mb=1420608; v=vector(mb); mx=mb^2; for(i=1, mm-1, for(j=i+1, mm, d=cb[j]-cb[i]; if(d<=mx, if(issquare(d, &r), v[r]=1), next(2)))); c=0; for(n=1, mb, if(v[n]==1, c++; write("b038596.txt", c " " n^2))) \\ Donovan Johnson, Oct 31 2013

a(n) = A038597(n)^2. - M. F. Hasler, Oct 05 2013

More terms from Jud McCranie

A038598 First differences between numbers that are a difference between 2 positive cubes.

Original entry on oeis.org

7, 12, 7, 11, 19, 5, 2, 28, 7, 19, 7, 3, 25, 17, 20, 19, 7, 2, 1, 53, 8, 17, 20, 15, 4, 7, 44, 1, 10, 51, 21, 16, 3, 16, 7, 2, 34, 55, 2, 27, 26, 8, 37, 19, 7, 56, 33, 2, 47, 9, 44, 17, 37, 15, 4, 7, 17, 11, 88, 26, 37, 19, 9, 10, 45, 6, 37, 19, 7, 22, 33, 2, 26, 55, 44, 7, 19, 65, 44, 10, 7

Offset: 1

Jeff Burch

nonn

Cf. A038593-A038597, A093330.

First differences of A181123.

Extended by Ray Chandler, Nov 29 2008

Offset corrected and a(1)=7 inserted by Sean A. Irvine, Jan 23 2021

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

A228946 Numbers m such that m^3 - k^3 is a square for some k < m, k > 0.

Original entry on oeis.org

8, 10, 14, 28, 32, 33, 40, 56, 57, 65, 71, 72, 74, 78, 90, 105, 112, 114, 126, 128, 130, 132, 140, 148, 154, 155, 160, 176, 193, 200, 217, 218, 224, 228, 250, 252, 260, 266, 273, 280, 284, 288, 296, 297, 305, 312, 329, 336, 344, 349, 350, 360, 392

Offset: 1

M. F. Hasler, Oct 05 2013

nonn

See A038596 = A038597^2 for the possible values of n^3-k^3.

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A001922, A038593-A038598, A014439, A014440, A014441, A066648, A071954.

isA228946 := proc (n) local k; for k to n-1 do if issqr(n^3-k^3) then return true end if end do; return false end proc; for n from 1 to 400 do if isA228946(n)then print(n): end if: end do: # Nathaniel Johnston, Oct 07 2013

k3sQ[n_]:=Count[n^3-Range[n-1]^3,?(IntegerQ[Sqrt[#]]&)]>0; Select[ Range[ 400],k3sQ] (* _Harvey P. Dale, May 07 2017 *)

for(n=1,999,for(k=1,n-1, issquare(n^3-k^3) & !print1(n",") & break))

is(n)=for(k=1,n-1, issquare(n^3-k^3) & return(1))

a(n) = A066648(n)^(1/3). - Amiram Eldar, Mar 20 2025

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

cp's OEIS Frontend

A038593 Differences between positive cubes in 1, 2 or 3 ways: union of A014439, A014440 and A014441.

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A181123 Numbers that are the differences of two positive cubes.

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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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A106265 Numbers a > 0 such that the Diophantine equation a + b^2 = c^3 has integer solutions b and c.

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A038596 Squares that are a difference between 2 positive cubes.

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A038598 First differences between numbers that are a difference between 2 positive cubes.

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

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