A024975 -id:A024975 - cp's OEIS frontend

A025399 Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.

36, 73, 92, 99, 134, 153, 160, 190, 197, 216, 225, 244, 251, 281, 288, 307, 342, 349, 352, 368, 371, 378, 405, 408, 415, 434, 469, 476, 495, 521, 532, 540, 547, 560, 567, 577, 584, 586, 603, 623, 638, 645, 664, 684, 701, 729, 736, 738, 755, 757, 764, 792, 794, 801, 820

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A025395 (not necessarily distinct), A024973.

Reap[For[n = 1, n <= 1000, n++, pr = Select[ PowersRepresentations[n, 3, 3], Times @@ # != 0 && Length[#] == Length[Union[#]] &]; If[pr != {} && Length[pr] == 1, Print[n, pr]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jul 31 2013 *)

A024975 MINUS A024974. - R. J. Mathar, May 28 2008

{n: A025469(n) = 1}. - R. J. Mathar, Jun 15 2018

A137365 Prime numbers n such that n = p1^3 + p2^3 + p3^3, a sum of cubes of 3 distinct prime numbers.

Original entry on oeis.org

1483, 5381, 6271, 7229, 9181, 11897, 13103, 13841, 14489, 17107, 20357, 25747, 26711, 27917, 30161, 30259, 31247, 32579, 36161, 36583, 36677, 36899, 36901, 42083, 48817, 54181, 55511, 55691, 56377, 56897, 57637, 59093, 64151, 66347

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 09 2008

nonn

Numbers n may have multiple decompositions; for example, n=185527 and n=451837 have two, and n=8627527 and n=32816503 have three. The smallest n with more than one decomposition is n = 185527 = 13^3+43^3+47^3 = 19^3+31^3+53^3, the 94th in the sequence. - R. J. Mathar, May 01 2008
Primes in A138853 and A138854. - M. F. Hasler, Apr 13 2008
The least prime, p, which has n decompositions {with its primes} is 1483 = {3, 5, 11}; 185527 = {13, 43, 47} & {19, 31, 53}; 8627527 = {19, 151, 173}, {33, 139, 181} & {71, 73, 199} and 1122871751 = {113, 751, 887}, {131, 701, 919}, {151, 659, 941} & {29, 107, 1039}. - Robert G. Wilson v, May 04 2008
The number of terms < 10^n: 0, 0, 0, 5, 56, 327, 2172, 13417, 86264, 567211, ..., . - Robert G. Wilson v, May 04 2008
The number of decompositions < 10^n: 0, 0, 0, 5, 56, 330, 2201, 13609, 87200, 571770, ..., . - Robert G. Wilson v, May 04 2008

			1483=3^3+5^3+11^3, 5381=17^3+7^3+5^3, 6271=3^3+11^3+17^3, etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..13418 (duplicates omitted)
Robert G. Wilson v, Table of n, a(n) for n = 1..13610 (duplicates included)
Index to sequences related to sums of cubes.

Cf. A137366.

Cf. A024975 (a^3+b^3+c^3, a>b>c>0), A122723 (primes in A024975), A138853-A138854.

# From R. J. Mathar: (Start)
isA030078 := proc(n) local cbr; cbr := floor(root[3](n)) ; if cbr^3 = n and isprime(cbr) then true ; else false; fi ; end:
isA137365 := proc(n) local p1,p2,p3,p3cub ; if isprime(n) then p1 := 2 ; while p1^3 <= n-16 do p2 := nextprime(p1) ; while p1^3+p2^3 <= n-8 do p3cub := n-p1^3-p2^3 ; if p3cub> p2^3 and isA030078(p3cub) then RETURN(true) ; fi ; p2 := nextprime(p2) ; od: p1 := nextprime(p1) ; od; RETURN(false) ; else RETURN(false) ; fi ; end:
for i from 1 do if isA137365( ithprime(i)) then printf("%d\n",ithprime(i)) ; fi ; od:
# (End)

Array[r, 99]; Array[y, 99]; For[i = 0, i < 10^2, r[i] = y[i] = 0; i++ ]; z = 4^2; n = 0; For[i1 = 1, i1 < z, a = Prime[i1]; a2 = a^3; For[i2 = i1 + 1, i2 < z, b = Prime[i2]; b2 = b^3; For[i3 = i2 + 1, i3 < z, c = Prime[i3]; c2 = c^3; p = a2 + b2 + c2; If[PrimeQ[p], Print[a2, " + ", b2, " + ", c2, " = ", p]; n++; r[n] = p]; i3++ ]; i2++ ]; i1++ ]; Sort[Array[r, 88]] (* Vladimir Joseph Stephan Orlovsky *)
lst = {}; Do[p = Prime[q]^3 + Prime[r]^3 + Prime[s]^3; If[PrimeQ@ p, AppendTo[lst, p]], {q, 13}, {r, q - 1}, {s, r - 1}]; Take[Sort@ lst, 36] (* Robert G. Wilson v, Apr 13 2008 *)
nn=20; lim=Prime[nn]^3+3^3+5^3; Union[Select[Total[#^3]& /@ Subsets[Prime[Range[2,nn]], {3}], #Harvey P. Dale, Jan 15 2011 *)

c=0; forprime(p=1,10^6, isA138853(p) & write("b137365.txt",c++," ",p)) \\ M. F. Hasler, Apr 13 2008

A137365 = A000040 intersect A138853 = A000040 intersect A138854. - M. F. Hasler, Apr 13 2008

Corrected and extended by Zak Seidov, R. J. Mathar and Robert G. Wilson v, Apr 12 2008

Further edits by R. J. Mathar and N. J. A. Sloane, Jun 07 2008

A306213 Numbers that are the sum of cubes of three distinct positive integers in arithmetic progression.

Original entry on oeis.org

36, 99, 153, 216, 288, 405, 408, 495, 645, 684, 792, 855, 972, 1071, 1197, 1224, 1407, 1548, 1584, 1701, 1728, 1968, 2079, 2241, 2304, 2403, 2541, 2673, 2736, 3051, 3060, 3240, 3264, 3537, 3540, 3888, 3960, 4059, 4131, 4257, 4500, 4587, 4833, 5049, 5160, 5256, 5472, 5643, 5832, 5940, 6336, 6369, 6669

Offset: 1

Antonio Roldán, Jan 29 2019

nonn

Numbers that can be written as 3*a*(a^2 + 2*b^2) = (a-b)^3 + a^3 + (a+b)^3 where 0 < b < a. - Robert Israel, Dec 15 2022

			153 = 1^3 + 3^3 + 5^3, with 3 - 1 = 5 - 3 = 2;
972 = 3^3 + 6^3 + 9^3, with 6 - 3 = 9 - 6 = 3.

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

Cf. A000578, A023042, A024975, A122723, A306212, A306214.

N:= 10000: # for terms <= N
S:= {}:
for a from 1 while a^3 + (a+1)^3 + (a+2)^3 <= N do
  for d from 1 do
    x:= a^3 + (a+d)^3 + (a+2*d)^3;
    if x > N then break fi;
    S:= S union {x}
od od:
sort(convert(S,list)); # Robert Israel, Dec 14 2022

for(n=3, 7000, k=(n/3)^(1/3); a=2; v=0; while(a<=k&&v==0, b=(n-3*a^3)/(6*a); if(b==truncate(b)&&issquare(b), d=sqrt(b), d=0); if(d>=1&&d<=a-1, v=1; print1(n,", ")); a+=1))

w=List(); for(n=3, 7000, k=(n/3)^(1/3); for(a=2, k, for(c=1, a-1, v=(a-c)^3+a^3+(a+c)^3; if(v==n, listput(w,n))))); print(vecsort(Vec(w),,8))

A138853 Numbers which are the sum of 3 cubes of distinct odd primes.

Original entry on oeis.org

495, 1483, 1701, 1799, 2349, 2567, 2665, 3555, 3653, 3871, 5065, 5283, 5381, 6271, 6369, 6587, 7011, 7137, 7229, 7235, 7327, 7453, 8217, 8315, 8441, 8533, 9083, 9181, 9399, 10387, 11799, 11897, 12115, 12319, 12537, 12635, 13103, 13525, 13623, 13841

Offset: 1

M. F. Hasler, Apr 13 2008

nonn

Dropping the restriction to odd primes would add to this sequence of odd terms the sequence of even terms of the form 8+p(i)^3+p(j)^3 (i>j>1), i.e. 8+{ even terms of A120398 }, cf. A138854.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..560 from R. J. Mathar)
Index to sequences related to sums of cubes.

Cf. A024975 (a^3+b^3+c^3, a>b>c>0), A138854, A120398.

isA138853(n)= local( c,d); n>494 && forprime( p=floor( sqrtn( n\3+1,3))+1, floor( sqrtn( n-151,3)), d=n-p^3; forprime( q=floor( sqrtn( d\2+1,3))+1, min( p-1, floor( sqrtn( d-26,3))), round( sqrtn( c=d-q^3,3 ))^3==c || next; isprime( round( sqrtn( c,3 ))) && return(1)))
forstep(n=3^3+5^3+7^3,10^5,2, isA138853(n)&print1(n", "))

A138853={ p(i)^3+p(j)^3+p(k)^3 ; i>j>k>1 }

A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

6, 188, 768, 1728, 2640, 21120, 42336, 13824, 71280, 5832, 80352, 74088, 425088, 421875, 1058400, 110592, 287496, 46656

Offset: 1

Zhining Yang, Jun 26 2025

a(19) > 2000000, a(20) = 216000, a(22) = 884736.

			a(3)=768, because 768^2 = 54^3 + 59^3 + 61^3 = 40^3 + 62^3 + 66^3 = 24^3 + 40^3 + 80^3 and no integer less than 768 has 3 solutions.

Cf. A024975, A025419, A377444.

s = Table[{k, Length@Select[PowersRepresentations[k^2, 3, 3],
     0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 2000}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 4}][[All, 1]]

a(18) from Chai Wah Wu, Jul 05 2025

A165454 Numbers the squares of which are sums of three distinct positive cubes.

Original entry on oeis.org

6, 15, 27, 48, 53, 59, 71, 78, 84, 87, 90, 96, 98, 116, 120, 121, 125, 134, 153, 162, 163, 167, 180, 188, 204, 213, 216, 224, 225, 226, 230, 240, 242, 244, 251, 253, 255, 262, 264, 280, 287, 288, 303, 314, 324, 330, 342, 350, 356, 363, 368, 372, 381, 384, 393

Offset: 1

Vladimir Joseph Stephan Orlovsky, Sep 20 2009

nonn

			6 is in the sequence because 6^2 = 1^3+2^3+3^3.
15 is in the sequence because 15^2 = 1^3+2^3+6^3.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A161992, A024973, A025399

N:= 1000: # to get all terms <= N
sc:= {seq(seq(seq(a^3 + b^3 + c^3, a = 1 .. min(b-1, floor((N^2 - b^3 - c^3)^(1/3)))), b = 2 .. min(c-1,floor((N^2 - c^3)^(1/3)))), c = 3 .. floor(N^(2/3)))}:
select(t -> member(t^2,sc), [$1..N]); # Robert Israel, Jan 27 2015

lst={};Do[Do[Do[d=Sqrt[a^3+b^3+c^3];If[d<=834&&IntegerQ[d],AppendTo[lst, d]],{c,b+1,5!,1}],{b,a+1,5!,1}],{a,5!}];Take[Union@lst,123]
Sqrt[# ]&/@Select[Total/@Subsets[Range[50]^3,{3}],IntegerQ[Sqrt[#]]&]// Union (* Harvey P. Dale, Oct 14 2020 *)

{k >0: k^2 in A024975}. [R. J. Mathar, Oct 06 2009]

Comments moved to the examples by R. J. Mathar, Oct 07 2009

Title corrected by Jeppe Stig Nielsen, Jan 26 2015

A327586 Numbers k such that k^4 = a^3 + b^3 + c^3 for some pairwise coprime positive integers a,b,c.

Original entry on oeis.org

39, 57, 70, 74, 106, 111, 147, 174, 209, 216, 236, 237, 244, 252, 291, 300, 318, 327, 333, 336, 342, 360, 366, 372, 387, 403, 417, 424, 450, 462, 489, 524, 540, 561, 582, 594, 615, 624, 636, 638, 651, 660, 673, 696, 700, 714, 739, 741, 768, 771, 804, 827, 837

Offset: 1

Robert Israel, Mar 03 2020

nonn

a(10) = 216 is the least term whose fourth power has two representations as a sum of the cubes of three pairwise coprime positive integers: 216^4 = 1217^3 + 639^3 + 484^3 = 1257^3 + 575^3 + 82^3. - Rémy Sigrist, Mar 04 2020

The least terms with 3 and 4 representations are a(230)=4914 and a(269)=5832, respectively. - Giovanni Resta, Mar 04 2020

			a(3) = 70 is a term because 70^4 = 81^3 + 167^3 + 266^3, and 81, 167 and 266 are positive and pairwise coprime.

Giovanni Resta, Table of n, a(n) for n = 1..549 (terms < 12000)
Mathematics StackExchange, Sum of three perfect cubes is equal to a perfect fourth
Giovanni Resta, Representations of k^4 for k<12000

Cf. A024975.

N:= 200: # to get all terms <= N
qmax:= N^4: Res:= {}:
for a from 1 while a^3 < qmax do
  for b from a+1 while a^3 + b^3 < qmax do
    if igcd(a,b) <> 1 then next fi;
    for c from b+1 while a^3 + b^3 + c^3 <= qmax do
      if igcd(c,a*b) <> 1 then next fi;
        q:= a^3 + b^3 + c^3;
        if issqr(q) and issqr(sqrt(q)) then
        Res:= Res union  {sqrt(sqrt(q))};
      fi
od od od:
sort(convert(Res,list));

More terms from Rémy Sigrist, Mar 04 2020

A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

3, 6, 16, 12, 27, 63, 38, 24, 94, 18, 123, 42, 93, 75, 141, 48, 66, 36, 153, 60, 140, 96, 279, 114, 200, 138, 410, 174, 72, 126, 186, 168, 204, 150, 108, 426, 132, 220, 418, 246, 498, 736, 144, 120, 294, 306, 210, 666, 282, 378, 252, 770, 216, 460, 462, 534, 180

Offset: 1

Zhining Yang, Jul 03 2025

nonn

			a(3)=16, because 16^6 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3  and no integer less than 16 has 3 solutions.

Chai Wah Wu, Table of n, a(n) for n = 1..104

Cf. A024975, A025419, A377444, A385354, A385565.

s = Table[{k, Length@Select[PowersRepresentations[k^6, 3, 3], 0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 30}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(41)-a(57) from Chai Wah Wu, Jul 07 2025

A112474 Squares that are the sum of three distinct positive cubes.

Original entry on oeis.org

36, 225, 729, 2304, 2809, 3481, 5041, 6084, 7056, 7569, 8100, 9216, 9604, 13456, 14400, 14641, 15625, 17956, 23409, 26244, 26569, 27889, 32400, 35344, 41616, 45369, 46656, 50176, 50625, 51076, 52900, 57600, 58564, 59536, 63001, 64009

Offset: 1

Rick L. Shepherd, Sep 06 2005

nonn

			36 = 6^2 = 1^3 + 2^3 + 3^3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A000290 and A024975.

Lim=64009;sqlim=Floor[Sqrt[Lim]];cblim=Ceiling[Lim^(1/3)]; Select[Range[sqlim]^2,MemberQ[ Union[Total/@Subsets[Range[cblim]^3,{3}]],#]&] (* James C. McMahon, Jun 04 2024 *)

has(n)=my(x3,z); for(x=sqrtnint(n\3,3)+1, sqrtnint(n,3), x3=x^3; for(y=sqrtnint((n-x3)\2,3)+1, min(x-1,sqrtnint(n-x3,3)), if(ispower(n-x3-y^3,3,&z) && z0, return(1)))); 0
list(lim)=my(v=List(),t); for(n=6,sqrtint(lim\1), if(has(t=n^2), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Sep 20 2016

Offset corrected by Charles R Greathouse IV, Sep 20 2016

A385565 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^4, where 0 < x < y < z has exactly n integer solutions.

Original entry on oeis.org

11, 21, 64, 144, 330, 846, 342, 252, 1331, 1008, 720, 1890, 3780, 729, 4200, 2016, 1000, 216, 6300, 8352, 10800, 12312, 8568, 19440, 8280, 9576, 21204

Offset: 1

Zhining Yang, Jul 03 2025

a(13) and a(15) not found up to k = 3300, a(14) = 729, a(16) = 2016, a(17) = 1000, a(18) = 216.

			a(3)=64, because 64^4 = 9^3 + 58^3 + 255^3 = 9^3 + 183^3 + 220^3 = 22^3 + 57^3 + 255^3 and no integer less than 64 has 3 solutions.

Cf. A024975, A025419, A377444, A385354, A385566.

s = Table[{k, Length@Select[PowersRepresentations[k^4, 3, 3],
      0 < #[[1]] < #[[2]] < #[[3]] &]}, {k, 400}];
a = Table[SelectFirst[s, #[[2]] == k &], {k, 5}][[All, 1]]

a(13), a(15), a(19)-a(21) from Chai Wah Wu, Jul 08 2025

a(22)-a(27) from Chai Wah Wu, Jul 18 2025

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A025399 Numbers that are the sum of 3 distinct positive cubes in exactly 1 way.

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A137365 Prime numbers n such that n = p1^3 + p2^3 + p3^3, a sum of cubes of 3 distinct prime numbers.

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A306213 Numbers that are the sum of cubes of three distinct positive integers in arithmetic progression.

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A138853 Numbers which are the sum of 3 cubes of distinct odd primes.

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A385354 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^2, where 0 < x < y < z has exactly n integer solutions.

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A165454 Numbers the squares of which are sums of three distinct positive cubes.

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A327586 Numbers k such that k^4 = a^3 + b^3 + c^3 for some pairwise coprime positive integers a,b,c.

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A385566 a(n) is the smallest positive integer k such that the Diophantine equation x^3 + y^3 + z^3 = k^6, where 0 < x < y < z has exactly n integer solutions.