A038593 -id:A038593 - cp's OEIS frontend

A038637 Numbers k such that k-th and (k+1)-st term of A038593 differ by 6.

65, 251, 253, 258, 349, 518, 537, 715, 860, 932, 934, 997, 1222, 1406, 1465, 1761, 1829, 2128, 2494, 2968, 2974, 2990, 3063, 3307, 3448, 3569, 4113, 4459, 5142, 5334, 5561, 6079, 6309, 6835, 7297, 7308, 7723, 8294, 8325, 8424, 8536, 8904, 9547, 9997, 10959, 11082

Offset: 1

Jeff Burch

nonn

Cf. A038651, A038652.

More terms from Jinyuan Wang, Mar 16 2020

A038662 First gap of n in sequence A038593 (upper terms).

Original entry on oeis.org

218, 63, 127, 335, 61, 1267, 26, 279, 875, 397, 37, 19, 21944, 17512, 331, 485, 169, 6813, 56, 189, 469, 1352, 63511, 3904, 152, 657, 631, 91, 2044, 1981, 55424, 2680, 817, 547, 2107, 2232, 702, 10305, 5528, 85736, 48769, 10565, 2906, 386, 1261, 4447

Offset: 1

Jeff Burch

nonn

A093362 Increasing gaps in A038593 (upper terms).

Original entry on oeis.org

19, 56, 91, 271, 602, 784, 1115, 2368, 3752, 4348, 6130, 7254, 13084, 20862, 22904, 28415, 56456, 57547, 75923, 92232, 201970, 268793, 306007, 413532, 422910, 497666, 918729, 1671130, 1710424, 2633408, 3907267, 5663644, 6660144, 6772519

Offset: 0

Jeff Burch, May 11 2004

nonn

Cf. A038593.

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

A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 13, 13, 1, 1, 19, 25, 19, 1, 1, 25, 37, 37, 25, 1, 1, 31, 49, 55, 49, 31, 1, 1, 37, 61, 73, 73, 61, 37, 1, 1, 43, 73, 91, 97, 91, 73, 43, 1, 1, 49, 85, 109, 121, 121, 109, 85, 49, 1, 1, 55, 97, 127, 145, 151, 145, 127, 97, 55, 1, 1, 61, 109, 145, 169, 181, 181, 169, 145, 109, 61, 1

Offset: 0

Kolosov Petro, Aug 31 2017

From Kolosov Petro, Apr 12 2020: (Start)
Let A(m, r) = A302971(m, r) / A304042(m, r).
Let L(m, n, k) = Sum_{r=0..m} A(m, r) * k^r * (n - k)^r.
Then T(n, k) = L(1, n, k), i.e T(n, k) is partial case of L(m, n, k) for m = 1.
T(n, k) is symmetric: T(n, k) = T(n, n-k). (End)

			Triangle begins:
  ----------------------------------------
  k=    0   1   2   3   4   5   6   7   8
  ----------------------------------------
  n=0:  1;
  n=1:  1,  1;
  n=2:  1,  7,  1;
  n=3:  1, 13, 13,  1;
  n=4:  1, 19, 25, 19,  1;
  n=5:  1, 25, 37, 37, 25,  1;
  n=6:  1, 31, 49, 55, 49, 31,  1;
  n=7:  1, 37, 61, 73, 73, 61, 37,  1;
  n=8:  1, 43, 73, 91, 97, 91, 73, 43,  1;

Georg Fischer, Table of n, a(n) for n = 0..495 [rows 0..10 and 12..30 from Kolosov Petro]
Petro Kolosov, On the link between binomial theorem and discrete convolution, arXiv:1603.02468 [math.NT], 2016-2025.
Petro Kolosov, Polynomial identity involving binomial theorem and Faulhaber's formula, 2023.
Petro Kolosov, History and overview of the polynomial P_b^m(x), 2024.

Columns k=0..6 give A000012, A016921, A017533, A161705, A103214, A128470, A158065.
Column sums k=0..4 give A000027, A000567, A051866, A051872, A255185.
Row sums give A001093.
Various cases of L(m, n, k): This sequence (m=1), A300656(m=2), A300785(m=3). See comments for L(m, n, k).
Differences of cubes n^3 are T(A000124(n), 1).
Cf. A000124, A000578, A003154, A003215, A007318, A008458, A016969.
Cf. A038593, A055012, A068601, A077028, A094053, A166873, A227776.
Cf. A294317, A302971, A304042.

Flat(List([0..11],n->List([0..n],k->6*k*(n-k)+1))); # Muniru A Asiru, Oct 09 2018

/* As triangle */ [[6*k*(n-k) + 1: k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 26 2018

T := (n, k) -> 6*k*(n-k) + 1:
seq(seq(T(n, k), k=0..n), n=0..11); # Muniru A Asiru, Oct 09 2018

T[n_, k_] := 6 k (n - k) + 1; Column[Table[T[n, k], {n, 0, 10}, {k, 0, n}], Center] (* Kolosov Petro, Jun 02 2019 *)

t(n, k) = 6*k*(n-k)+1
trianglerows(n) = for(x=0, n-1, for(y=0, x, print1(t(x, y), ", ")); print(""))
/* Print initial 9 rows of triangle as follows */
trianglerows(9) \\ Felix Fröhlich, Jan 09 2018

def A287326(n,k): return 6*k*(n-k) + 1
flatten([[A287326(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 25 2024

T(n, k) = 6*k*(n-k) + 1.
G.f. of column k: n^k*(1+(6*k-1)*n)/(1-n)^2.
G.f.: (1 - x - x*y + 7*x^2*y)/((1 - x)^2*(1 - x*y)^2). - Stefano Spezia, Oct 09 2018 [Adapted by Stefano Spezia, Sep 25 2024]
From Kolosov Petro, Jun 05 2019: (Start)
T(n, k) = 1/2 * T(A294317(n, k), k) + 1/2.
T(n+1, k) = 2*T(n, k) - T(n-1, k), for n >= k.
T(n, k) = 6*A077028(n, k) - 5.
T(2n, n) = A227776(n).
T(2n+1, n) = A003154(n+1).
T(2n+3, n) = A166873(n+1).
Sum_{k=0..n-1} T(n, k) = Sum_{k=1..n} T(n, k) = A000578(n).
Sum_{k=1..n-1} T(n, k) = A068601(n).
(n+1)^3 - n^3 = T(A000124(n), 1). (End)
Sum_{k=0..n} (-1)^k*T(n, k) = (-1/2)*(1 + (-1)^n)*A016969(floor(n/2) - 1). - G. C. Greubel, Sep 25 2024

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

A014439 Differences between two positive cubes in exactly 1 way.

Original entry on oeis.org

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, 784, 817, 819, 866, 875, 919, 936, 973, 988, 992

Offset: 1

Glen Burch (gburch(AT)erols.com)

nonn

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

Cf. A000578, A038593, A181123, A034179 (more than one way), A014440 (exactly two ways), A265625 (more than two ways), A014441 (exactly three ways), A333376, A333377.

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] = 1, [$1..N]); # Robert Israel, Dec 11 2015

r = 992; 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]] == 1 &], {2, -1, 2}] (* Arkadiusz Wesolowski, Dec 10 2015 *)

Corrected and extended by Don Reble, Nov 19 2006

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

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

cp's OEIS Frontend

A038637 Numbers k such that k-th and (k+1)-st term of A038593 differ by 6.

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A038662 First gap of n in sequence A038593 (upper terms).

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A093362 Increasing gaps in A038593 (upper terms).

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A181123 Numbers that are the differences of two positive cubes.

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A287326 Triangle read by rows: T(n, k) = 6*k*(n-k) + 1; n >= 0, 0 <= k <= n.

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

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A014439 Differences between two positive cubes in exactly 1 way.

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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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A287326 Triangle read by rows: T(n, k) = 6k(n-k) + 1; n >= 0, 0 <= k <= n.