cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

T. D. Noe, Oct 06 2010

Keywords

Comments

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

Links

Crossrefs

Cf. A014439, A014440, A014441, A333376, A333377, A038593 (subsequence), A086120.
Cf. A024352 (squares), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

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

Original entry on oeis.org

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

Views

Author

Hollie L. Buchanan II, Jul 11 2003

Keywords

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 &]

Extensions

More terms from Alexander Adamchuk, Nov 10 2006

A086121 Positive sums or differences of two cubes of primes.

Original entry on oeis.org

16, 19, 35, 54, 98, 117, 133, 152, 218, 250, 316, 335, 351, 370, 468, 686, 866, 988, 1206, 1304, 1323, 1339, 1358, 1456, 1674, 1854, 1946, 2072, 2170, 2189, 2205, 2224, 2322, 2540, 2662, 2716, 3528, 3582, 4394, 4570, 4662, 4788, 4886, 4905, 4921, 4940, 5038
Offset: 1

Views

Author

Hollie L. Buchanan II, Jul 11 2003

Keywords

Examples

			117 and 133 each belong to the (set) sequence because can be written as 117 = 5^3 - 2^3 and 133 = 5^3 + 2^3.

Links

Crossrefs

Cf. A086119, A086120. Also see A045636, A045699.

Programs

  • Mathematica
    nn=10^6; td=Reap[Do[n=Prime[i]^3-Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[Sqrt[nn/6]]}, {j,i-1}]][[2,1]]; ts=Reap[Do[n=Prime[i]^3+Prime[j]^3; If[n<=nn, Sow[n]], {i,PrimePi[nn^(1/3)]}, {j,i}]][[2,1]]; Union[td,ts] (* T. D. Noe, Oct 04 2010 *)
n = 100; Select[Sort@Flatten@ Table[Prime[i]^3 + (-1)^k Prime[j]^3, {i, n}, {j, i}, {k, 2}], 0 < # < (Prime[n] + 2)^3 - Prime[n]^3 &] (* Ray Chandler, Oct 05 2010 *)

Extensions

Edited by N. J. A. Sloane, Oct 05 2010 to remove a discrepancy between the terms of the sequence and the b-file. The old Mma program and b-file were wrong.
Showing 1-3 of 3 results.

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