A038596 -id:A038596 - 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

A038597 Numbers whose square is a difference between 2 positive cubes in at least one way.

Original entry on oeis.org

13, 28, 49, 104, 147, 181, 189, 224, 351, 361, 388, 392, 507, 549, 588, 676, 756, 832, 1029, 1176, 1323, 1369, 1425, 1448, 1512, 1625, 1792, 1862, 1911, 1922, 2299, 2355, 2521, 2808, 2883, 2888, 3104, 3136, 3185, 3216, 3500, 3721, 3969, 4056, 4103, 4332

Offset: 1

Jeff Burch

nonn

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

Cf. A038593-A038598.

is(n)=my(N=n^2); for(k=sqrtnint(N,3)+1,(sqrtint(12*N-3)+3)\6, if(ispower(N-k^3,3), return(1))); 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("b038597.txt", c " " n))) \\ Donovan Johnson, Oct 31 2013

a(n) = sqrt(A038596(n)). - M. F. Hasler, Oct 05 2013

More terms from Jud McCranie

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

A051393 Numbers whose square is expressible as the difference of positive cubes in more than one way.

Original entry on oeis.org

36963, 66248, 157339, 262808, 295704, 519841, 529984, 793117, 990584, 998001, 1258712, 1271403, 1291836, 1788696, 2102464, 2365632, 2688728, 3294486, 3756536, 4158728, 4239872, 4248153, 4620375, 5532247, 5738904, 5938947, 6344936, 6562101, 7095816, 7924672

Offset: 1

Jud McCranie

nonn

			36963^2 = 1110^3-111^3 = 1332^3-999^3.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A038596, A038597.

mm=577350; cb=vector(mm); for(i=1, mm, cb[i]=i^3); v=vector(10^6); n=vector(10); c=0; mx=10^12; for(i=1, mm-1, for(j=i+1, mm, s=cb[j]-cb[i]; if(s<=mx, if(issquare(s,&r), v[r]++; if(v[r]==2, c++; n[c]=r)), next(2)))); n=vecsort(n); for(i=1, c, print1(n[i] ", ")) \\ Donovan Johnson, Oct 29 2013

Corrected and extended by Ray Chandler, Dec 01 2008

Offset corrected by Donovan Johnson, Oct 29 2013

A129965 Triangular numbers that are the difference of nonnegative cubes.

Original entry on oeis.org

0, 1, 91, 4095, 5886, 7875, 8128, 8911, 9045, 17955, 21736, 23653, 47278, 93961, 115921, 130816, 184528, 259560, 379756, 488566, 575128, 658378, 758296, 810901, 873181, 885115, 1060696, 1155960, 1358776, 1385280, 1997001, 2616328, 2685403

Offset: 1

Peter Pein (petsie(AT)dordos.net), Jun 13 2007, Jun 14 2007

nonn

			A000217(13) = Sum_{k=1..13} k = 91 = 216 - 125 = 6^3 - 5^3, so 91 is in the sequence. - _Peter Munn_, Dec 05 2022

Robert Israel, Table of n, a(n) for n = 1..439 (terms <= 10^12)

Intersection of A000217 and A152043.

Cf. A038596, A129966, A185253.

M:= 10^7: # for terms <= M
S:= {0}:
for x from 1 while 3*x^2 - 3*x + 1 < M do
   if x^3 < M then Y:= 0 else Y:= ceil(x^3-M) fi;
  S:= S union select(t -> issqr(1+8*t),{seq(x^3 - y^3, y = Y .. x-1)});
od:
sort(convert(S,list)); # Robert Israel, Dec 05 2023

With[{n = 5000}, Intersection[(#1*((#1 + 1)/2) & ) /@ Range[0, n], Flatten[Outer[ #1^3 - #2^3 &, Range[n], Range[0, n - 1]]]]]

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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A038597 Numbers whose square is a difference between 2 positive cubes in at least one way.

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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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A230716 Numbers whose square is both a sum and a difference of two positive cubes.

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A230717 Squares that are both a sum and a difference of two positive cubes.

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PARI

Formula

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A051393 Numbers whose square is expressible as the difference of positive cubes in more than one way.

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PARI

Extensions

A129965 Triangular numbers that are the difference of nonnegative cubes.

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Maple

Mathematica