A003325 -id:A003325 - cp's OEIS frontend

A086119 Numbers of the form p^3 + q^3, p, q primes.

16, 35, 54, 133, 152, 250, 351, 370, 468, 686, 1339, 1358, 1456, 1674, 2205, 2224, 2322, 2540, 2662, 3528, 4394, 4921, 4940, 5038, 5256, 6244, 6867, 6886, 6984, 7110, 7202, 8190, 9056, 9826, 11772, 12175, 12194, 12292, 12510, 13498, 13718, 14364

Offset: 1

Hollie L. Buchanan II, Jul 11 2003

nonn

			133 belongs to the sequence because it can be written as 2^3 + 5^3.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A001235, A003325, A045636, A045699, A086120, A086121.

sumList[x_List, y_List] := Module[{t = {}}, Do[t = Union[t, x[[i]] + y], {i, Length[x]}];  t]; nn = 10; Select[sumList[Prime[Range[nn]]^3, Prime[Range[nn]]^3], # < Prime[nn]^3 &]

More terms from Alexander Adamchuk, Nov 10 2006

A045980 Numbers of the form x^3 + y^3 or x^3 - y^3.

Original entry on oeis.org

0, 1, 2, 7, 8, 9, 16, 19, 26, 27, 28, 35, 37, 54, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 128, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 250, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 432

Offset: 1

N. J. A. Sloane

Sums of two integer cubes. - Charles R Greathouse IV, Mar 30 2022

			7 = (2)^3 + (-1)^3.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 86.

T. D. Noe, Table of n, a(n) for n = 1..1000
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
Kevin A. Broughan, Characterizing the sum of two cubes, J. Integer Seqs., Vol. 6, 2003.
M. Kim, Diophantine equations in two variables, arXiv:math/0210329 [math.NT], 2002.
Index entries for sequences related to sums of cubes

A004999 and A003325 are subsequences.

Cf. A222304, A222305, A222306, A027750, A010052.

a045980 n = a045980_list !! (n-1)
a045980_list = 0 : filter f [1..] where
   f x = g $ takeWhile ((<= 4 * x) . (^ 3)) $ a027750_row x where
     g [] = False
     g (d:ds) = r == 0 && a010052 (d ^ 2 - 4 * y) == 1 || g ds
       where (y, r) = divMod (d ^ 2 - div x d) 3
-- Reinhard Zumkeller, Dec 20 2013

Union[Select[Sort[Flatten[Table[{j^3-i^3, j^3+i^3}, {i, 0, 20}, {j, i, 20}]]], #<20^3-19^3&]]
With[{nn=20},Take[Union[Select[Flatten[{Total[#],#[[1]]-#[[2]]}&/@(Tuples[ Range[0,nn],2]^3)],#>-1&]],3*nn]] (* Harvey P. Dale, Jun 22 2014 *)

is(n)=fordiv(n,d, my(L=(d^2-n/d)/3); if(denominator(L)==1 && issquare(d^2-4*L), return(1))); 0 \\ Charles R Greathouse IV, Jun 12 2012

list(lim)={
    my(v=List(),x3,t);
    for(x=0,sqrtnint(lim\=1,3),
        x3=x^3;
        for(y=0,min(sqrtnint(lim-x3,3),x),
            listput(v,x3+y^3)
        )
    );
    for(x=2,t=sqrtint(lim\3),
        x3=x^3;
        for(y=sqrtnint(max(0,x3-lim-1),3)+1,x-1,
            listput(v,x3-y^3)
        )
    );
    t=(t+1)^3-t^3;
    if(t<=lim,listput(v,t));
    Set(v);
} \\ Charles R Greathouse IV, Jun 12 2012, updated Jan 13 2022

is(n)=#thue(thueinit(z^3+1),n) \\ Ralf Stephan, Oct 18 2013

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of ways to write n as an ordered sum of two positive cubes.

			a(9) = 2 because we have [8, 1] and [1, 8].

Antti Karttunen, Table of n, a(n) for n = 0..65537
Eric Weisstein's World of Mathematics, Cubic Number
Index entries for sequences related to sums of cubes

Cf. A000578, A001235 (positions of terms > 3), A003325 (of nonzero terms), A010057, A063725, A173677.

nmax = 150; CoefficientList[Series[(Sum[x^(k^3), {k, 1, nmax}])^2, {x, 0, nmax}], x]

A010057(n) = ispower(n, 3);
A280618(n) = if(n<2, 0, sum(r=1,sqrtnint(n-1,3),A010057(n-(r^3)))); \\ Antti Karttunen, Nov 30 2021

G.f.: (Sum_{k>=1} x^(k^3))^2.

A266230 Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

3, 1, 3703, 5, 43, 2, 119, 3, 1, 19, 5, 384, 2, 29, 29, 1, 7, 18, 6, 3, 13, 14, 869, 7, 2, 15, 3, 1, 10, 5, 23, 2, 20, 10, 1, 45, 6, 2373, 4, 1193, 5, 52, 7, 36, 54, 3, 18, 5, 13, 4, 2, 385, 9, 1, 14, 6, 3, 76, 250, 250, 34, 2, 8, 3, 1, 336, 5, 52, 2, 8, 28, 1, 21, 12, 13, 4, 113

Offset: 0

Zhi-Wei Sun, Dec 24 2015

nonn

Conjecture:  For any integer m, there are positive integers x, y and z such that m + x^2 = y^3 + z^3.
This is similar to the conjecture in A266152. We have verified it for all integers m with |m| <= 25000.
Obviously, a(k^3) = 1 for any positive integer k.
See also A266231 for a related sequence.

			a(0) = 3 since 0 + 3^2 = 1^3 + 2^3.
a(2) = 3703 since 2 + 3703^2 = 107^3 + 232^3.
a(3) = 5 since 3 + 5^2 = 1^3 + 3^3.
a(4) = 43 since 4 + 43^2 = 5^3 + 12^3.
a(37) = 2373 since 37 + 2373^2 = 93^3 + 169^3.
a(1227) = 132316 since 1227 + 132316^2 = 1874^3 + 2219^3.

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Checking the conjecture for integers m with 10000 < |m| <= 25000

Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266231.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=1;Label[bb];Do[If[CQ[n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,0,80}]

A266231 Least positive integer x such that x^2 - n = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Original entry on oeis.org

6, 2, 61, 47, 3283, 16, 3, 6, 5, 8, 12, 686, 16, 4, 302, 5, 13, 12, 152, 6, 7, 83, 5, 148, 33, 37, 6, 10, 8, 11, 34, 16, 7, 6, 10, 8, 24, 53, 16, 7, 13, 52, 13, 14, 30, 9, 7, 8, 11, 67, 74, 22, 9, 28, 8, 11, 43, 115, 20, 122, 23, 8, 14, 48, 9, 25, 11, 14, 392, 14

Offset: 1

Zhi-Wei Sun, Dec 24 2015

nonn

The conjecture in A266230 implies that a(n) > 0 for all n > 0.

			 a(1) = 6 since 6^2 - 1 = 2^3 + 3^3.
a(3) = 61 since 61^2 - 3 = 7^3 + 15^3.
a(4) = 47 since 47^2 - 4 = 2^3 + 13^3.
a(5) = 3283 since 3283^2 - 5 = 65^3 + 219^3.
a(166) = 6554 since 6554^2 - 166 = 175^3 + 335^3.
a(635) = 44779 since 44779^2 - 635 = 25^3 + 1261^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A003325, A266152, A266153, A266212, A266230.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[x=Floor[Sqrt[n]]+1;Label[bb];Do[If[CQ[-n+x^2-y^3],Print[n," ",x];Goto[aa]],{y,1,((-n+x^2)/2)^(1/3)}];x=x+1;Goto[bb];Label[aa];Continue,{n,1,70}]

A025455 a(n) is the number of partitions of n into 2 positive cubes.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

David W. Wilson

nonn

In other words, number of solutions to the equation x^3 + y^3 = n with x >= y > 0. - Antti Karttunen, Aug 28 2017

The first term > 1 is a(1729) = 2. - Michel Marcus, Apr 23 2019

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to sums of cubes

Cf. A025456, A025468, A003108, A003325, A000578, A048766, A001235 (two or more ways, positions where a(n) > 1).

Cf. also A025426, A216284.

Table[Count[IntegerPartitions[n,{2}],?(AllTrue[Surd[#,3],IntegerQ]&)],{n,0,110}] (* _Harvey P. Dale, Nov 23 2022 *)

(define (A025455 n) (let loop ((x (A048766 n)) (s 0)) (let* ((x3 (A000578 x)) (y3 (- n x3))) (if (< x3 y3) s (loop (- x 1) (+ s (if (and (> y3 0) (= (A000578 (A048766 y3)) y3)) 1 0))))))) ;; Antti Karttunen, Aug 28 2017

If a(n) > 0 then A025456(n + k^3) > 0 for k>0; a(A113958(n)) > 0; a(A003325(n)) > 0. - Reinhard Zumkeller, Jun 03 2006
a(n) >= A025468(n). - Antti Karttunen, Aug 28 2017
a(n) = [x^n y^2] Product_{k>=1} 1/(1 - y*x^(k^3)). - Ilya Gutkovskiy, Apr 23 2019

Secondary offset added by Antti Karttunen, Aug 28 2017

Secondary offset corrected by Michel Marcus, Apr 23 2019

A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

2, 9, 41, 202, 938, 4354, 20330, 94625, 439959, 2045048, 9500746, 44124084, 204883131, 951202028, 4415710979, 20497646229, 95146359635

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + a(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003325.

iscube:=proc(n) if root(n,3)=trunc(root(n,3)) then true; else false; fi; end:
isA003325:=proc(n) local x,y3; if iscube(n) then false; else for x from 1 do y3:=n-x^3; if y3A003325(k) then i:=i+1; fi; od: return(i); end:
for n from 1 do print(a(n)); od;

a(n)=my(N=10^n,v=List(),x3);sum(x=1,sqrtnint(N-1,3),x3=x^3;sum(y=1, min(sqrtnint(N-x3,3),x), !ispower(x3+y^3,3) && listput(v,x3+y^3))); #vecsort(v,,8) \\ Charles R Greathouse IV, Oct 16 2013

a(6)-a(12) from Lars Blomberg, May 04 2011

a(13)-a(17) from Hiroaki Yamanouchi, Jul 12 2014

A226903 Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.

Original entry on oeis.org

5, 18, 53, 102, 197, 306, 491, 684, 989, 1290, 1745, 2178, 2813, 3402, 4247, 5016, 6101, 7074, 8429, 9630, 11285, 12738, 14723, 16452, 18797, 20826, 23561, 25914, 29069, 31770, 35375, 38448, 42533, 46002, 50597, 54486, 59621, 63954, 69659, 74460, 80765, 86058

Offset: 1

Jonathan Sondow, Jun 22 2013

Shiraishi's solutions to a^3 + b^3 + c^3 = d^3 are a = 3n^2; b = 6n^2 - 3n + 1 or 6n^2 + 3n + 1; c = 9n^3 - 6n^2 + 3n - 1 or  9n^3 + 6n^2 + 3n, respectively, for n > 0; and d = c+1. See Smith and Mikami for a derivation.
Shiraishi's formulas show that the sequence is infinite. Hence the sequences A023042 (solutions to x^3 + y^3 + z^3 = w^3), A225908 (solutions to a^3 + b^3 = c^3 - d^3), A225909 (solutions to a^3 + b^3 = (c+1)^3 - c^3) and A226902 (numbers c in A225909) are also infinite.
Shiraishi's solution b = 6n^2 +/- 3n + 1 is the centered triangular numbers A005448 except 1.

			The first two terms are a(1) = 9 - 6 + 3 - 1 = 5 and a(2) = 9 + 6 + 3 = 18. Then Shiraishi's formulas give 3^3 + 4^3 + 5^3 = 6^3 and 3^3 + 10^3 + 18^3 = 19^3.

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
Wikipedia (French), Shiraishi Nagatada
Wikipedia (German), Shiraishi Nagatada
Index entries for sequences related to sums of cubes
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A003325, A005448, A023042, A181123, A225908, A225909, A226902.

a(2n-1) = 9n^3 - 6n^2 + 3n - 1.
a(2n) = 9n^3 + 6n^2 + 3n.
G.f.: x*(5 + 13*x + 20*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1 + x)^3*(1 - x)^4). [Bruno Berselli, Jun 22 2013]
a(n) = (18*n^3 + 27*n^2 + 27*n + 1 - (3*n^2 + 3*n + 1)*(-1)^n)/16. [Bruno Berselli, Jun 22 2013]
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 7. - Chai Wah Wu, Aug 05 2025

A181376 Total number of n-digit numbers requiring 2 positive cubes in their representation as a sum of cubes.

Original entry on oeis.org

2, 7, 32, 161, 736, 3416, 15976, 74295, 345334, 1605089, 7455698, 34623338, 160759047, 746318897, 3464508951, 16081935250, 74648713406

Offset: 1

Martin Renner, Jan 28 2011

A181354(n) + a(n) + A181378(n) + A181380(n) + A181384(n) + A181401(n) + A181403(n) + A181405(n) + A171386(n) = A052268(n).

			a(1) = 2 from 1+1=2, 1+8=9.
a(2) = 7 from 8+8=16, 1+27=28, 35, 54, 65, 72, 91.

Eric W. Weisstein, MathWorld -- Waring's Problem.

Cf. A003325.

Table[Length[c = Table[j^3, {j, (10^n - 1)^(1/3)}];
  Select[Union[Flatten[Outer[Plus, c, c]]],
IntervalMemberQ[Interval[{10^(n - 1), 10^n - 1}], #] &]], {n, 10}] (* Robert Price, Apr 18 2019 *)

a(n)=my(N=10^n, Nn=N/10, v=List(), x3, t); sum(x=sqrtnint(Nn\2,3), sqrtnint(N-1, 3), x3=x^3; sum(y=1, min(sqrtnint(N-x3, 3), x), t=x3+y^3; t>=Nn && !ispower(t, 3) && listput(v, t))); #vecsort(v, , 8) \\ Charles R Greathouse IV, Oct 16 2013

a(n) = A181375(n)-A181375(n-1).

a(6)-a(11) from Charles R Greathouse IV, Oct 16 2013
a(12) from Lars Blomberg, Jan 15 2014
a(13)-a(17) from Hiroaki Yamanouchi, Jul 13 2014

A101421 Numbers which are the sum of two positive cubes and divisible by 7.

Original entry on oeis.org

28, 35, 91, 126, 133, 189, 217, 224, 280, 539, 637, 686, 728, 756, 854, 945, 1001, 1008, 1064, 1358, 1456, 1512, 1547, 1729, 1736, 1792, 2198, 2205, 2240, 2261, 2331, 2457, 2709, 2926, 3059, 3087, 3402, 3500, 3528, 3591, 4123, 4221, 4312, 4375, 4914, 4921

Offset: 1

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 17 2005

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A003325.

upto[n_] := Block[{t}, Union@ Reap[Do[If[Mod[t = x^3 + y^3, 7] == 0, Sow@t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}]][[2, 1]]]; upto[5000] (* Giovanni Resta, Jun 12 2020 *)

Changed offset from 0 to 1 by Vincenzo Librandi, May 08 2013

cp's OEIS Frontend

A086119 Numbers of the form p^3 + q^3, p, q primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A045980 Numbers of the form x^3 + y^3 or x^3 - y^3.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

PARI

A280618 Expansion of (Sum_{k>=1} x^(k^3))^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A266230 Least positive integer x such that n + x^2 = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A266231 Least positive integer x such that x^2 - n = y^3 + z^3 for some positive integers y and z, or 0 if no such x exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A025455 a(n) is the number of partitions of n into 2 positive cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Scheme

Formula

Extensions

A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI