A014439 -id:A014439 - 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

A014440 Differences between two positive cubes in exactly 2 ways.

Original entry on oeis.org

721, 728, 999, 5768, 5824, 5859, 7992, 8911, 9919, 10621, 12663, 12824, 19467, 19656, 23877, 25669, 26973, 27937, 28063, 34209, 35208, 35929, 41743, 43561, 46144, 46592, 46872, 49959, 53144, 63936, 68857, 68913, 71288, 77779, 79352, 80379, 84968, 90125

Offset: 1

Glen Burch (gburch(AT)erols.com)

nonn

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A265625 (more than two ways), A014441 (exactly three ways), A333376, A333377.

r = 90125; 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]], #[[2]] == 2 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Extended by Don Reble, Nov 19 2006

A014441 Differences between two positive cubes in exactly 3 ways.

Original entry on oeis.org

3367, 26936, 90909, 152551, 205352, 215488, 420875, 622232, 625177, 727272, 754299, 757701, 845208, 1147627, 1154881, 1220408, 1642816, 1723904, 2113921, 2454543, 2741256, 3056473, 3367000, 3442887, 3492125, 4481477, 4977856, 5001416, 5544504, 5818176

Offset: 1

Glen Burch (gburch(AT)erols.com)

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A014440 (exactly two ways), A265625 (more than two ways), A333376 (exactly 4 ways), A333377 (exactly 5 ways).

r = 5818176; 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]], #[[2]] == 3 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Extended by Don Reble, Nov 19 2006

A333376 Differences between two positive cubes in exactly 4 ways.

Original entry on oeis.org

4118877, 32951016, 52324993, 94287375, 111209679, 214435711, 263608128, 418599944, 442245349, 514859625, 754299000, 889677432, 995635368, 1080305856, 1147627000, 1715485688, 2108865024, 2545759125, 3002661333, 3348799552, 3537962792, 3701994688, 4118877000, 4304670552

Offset: 1

Giovanni Resta, Mar 17 2020

			4118877 = 162^3 - 51^3 = 165^3 - 72^3 = 178^3 - 115^3 = 678^3 - 675^3.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333377, A098110.

Subsequence of A265625.

a(1) = A098110(4).

A333377 Differences between two positive cubes in exactly 5 ways.

Original entry on oeis.org

1412774811, 11302198488, 38144919897, 90417587904, 105443078832, 176596851375, 305159359176, 370544908608, 484581760173, 723340703232, 843544630656, 1029912837219, 1238805803151, 1412774811000, 1808088149952, 1880403273441, 2441274873408, 2846963128464, 2863636114248

Offset: 1

Giovanni Resta, Mar 17 2020

			1412774811 = 1134^3 - 357^3 = 1155^3 - 504^3 = 1246^3 - 805^3 = 2115^3 - 2004^3 = 4746^3 - 4725^3.

Giovanni Resta, Table of n, a(n) for n = 1..150

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A098110.

a(1) = A098110(5).

A034179 Difference between two positive cubes in more than one way.

Original entry on oeis.org

721, 728, 999, 3367, 5768, 5824, 5859, 7992, 8911, 9919, 10621, 12663, 12824, 19467, 19656, 23877, 25669, 26936, 26973, 27937, 28063, 34209, 35208, 35929, 41743, 43561, 46144, 46592, 46872, 49959, 53144, 63936, 68857, 68913, 71288, 77779

Offset: 1

Erich Friedman

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A014439, A014440, A014441, A333376, A333377, A181123, A265625.

fQ[n_] := Block[{r = Reduce[0 < y < x && n == x^3 - y^3, {x, y}, Integers]}, If[r === False || Head[r] === And, False, Length[r] >= 2]]; Select[ Range[77780], If[ fQ[#], Print[#]; True, False] &] (* Jean-François Alcover, Apr 11 2011 *)
With[{nn=50},Take[Sort[Transpose[Select[Tally[#[[2]]-#[[1]]&/@Subsets[ Range[ nn*20]^3,{2}]],#[[2]]>1&]][[1]]],nn]] (* Harvey P. Dale, Mar 09 2016 *)

Extended by Ray Chandler, Nov 29 2008

Offset corrected by Giovanni Resta, Mar 17 2020

A265625 Differences between two positive cubes in more than two ways.

Original entry on oeis.org

3367, 26936, 90909, 152551, 205352, 215488, 420875, 622232, 625177, 727272, 754299, 757701, 845208, 1147627, 1154881, 1220408, 1642816, 1723904, 2113921, 2454543, 2741256, 3056473, 3367000, 3442887, 3492125, 4118877, 4481477, 4977856, 5001416, 5544504

Offset: 1

Arkadiusz Wesolowski, Dec 10 2015

nonn

			3367 = 15^3 - 2^3 = 16^3 - 9^3 = 34^3 - 33^3.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000578, A181123, A014439 (exactly one way), A034179 (more than one way), A014440 (exactly two ways), A014441 (exactly three ways), A333376, A333377.

r = 5544504; p = 3; Rest@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]], #[[2]] > 2 &], {2, -1, 2}]

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

A098110 Smallest number that is the difference between two positive cubes in n ways.

Original entry on oeis.org

7, 721, 3367, 4118877, 1412774811, 424910390480793

Offset: 1

Jeff Burch, Sep 23 2004

a(7) <= 15490327057569000, a(8) <= 123922616460552000. - Giovanni Resta, Mar 19 2020

			Pairs (x, y) such that x^3 - y^3 = a(1), ..., a(6):
7 = (2, 1);
721 = (16, 15), (9, 2);
3367 = (34, 33), (16, 9), (15, 2)l
4118877 = (162, 51), (165, 72), (178, 115), (678, 675);
1412774811 = (1134, 357), (1155, 504), (1246, 805), (2115, 2004), (4746, 4725);
424910390480793 = (596001, 595602), (317982, 316575), (141705, 134268), (83482, 53935), (77385, 33768), (75978, 23919).

Cf. A014439, A014440, A014441, A181123, A034179, A265625, A333376, A333377.

Cf. A011541,A047696.

a(6) from Giovanni Resta, Mar 19 2020

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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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A014440 Differences between two positive cubes in exactly 2 ways.

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A014441 Differences between two positive cubes in exactly 3 ways.

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A333376 Differences between two positive cubes in exactly 4 ways.

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A333377 Differences between two positive cubes in exactly 5 ways.

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A034179 Difference between two positive cubes in more than one way.

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A265625 Differences between two positive cubes in more than two ways.

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

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