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.

A293648 Sum of two (possibly negative) coprime cubes in at least 2 ways, but not the sum of 2 noncoprime cubes.

Original entry on oeis.org

91, 217, 721, 1027, 1729, 3367, 4706, 4921, 4977, 7657, 8587, 8911, 9919, 10621, 14911, 15561, 16263, 20683, 21014, 23877, 25669, 27937, 28063, 31519, 35929, 39331, 40033, 49959, 63693, 68705, 68857, 68913, 73017, 77653, 77779, 97309, 98623, 106597, 109573
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Not every term is cubefree; some are sb^3 where s is in A159843 and b > 1.

Examples

			63693 = 34^3 + 29^3 = 53^3 - 44^3 and 34 & 29 are coprime, as are 53 & -44, but it is not also the sum of cubes of 2 noncoprime integers, so 63693 is in the sequence.

Links

Crossrefs

Cf. A293647 (allows noncoprime); A293649 (only positives); A293646, A293651

A293645 Positive numbers that are the sum of two (possibly negative) coprime cubes.

Original entry on oeis.org

1, 2, 7, 9, 19, 26, 28, 35, 37, 61, 63, 65, 91, 98, 117, 124, 126, 127, 133, 152, 169, 189, 215, 217, 218, 271, 279, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 468, 469, 485, 511, 513, 539, 547, 559, 602, 604, 631, 637, 657, 665, 721, 728, 730
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Also sum or difference of two coprime cubes. - David A. Corneth, Oct 20 2017

Examples

			19 = 3^3 + (-2)^3, where 3 and -2 are coprime, so 19 is in the sequence.
152 = 5^3 + 3^3, where 5 and 3 are coprime, so 152 is in the sequence.

Links

Crossrefs

Cf. A003325 (positive cubes); A020895 (cubefree); A293646 (only coprime); A293647, A293650.

Programs

  • Maple
    filter:= proc(n) local s,x,y;
  for s in numtheory:-divisors(n) do
    x:= s/2 + sqrt(12*n/s-3*s^2)/6;
    if not x::integer then next fi;
    y:= s - x;
    if igcd(x,y) = 1 then return true fi;
  od;
  false
end proc:
select(filter, [seq(seq(9*i+j,j=[1,2,7,8,9]),i=0..1000)]); # Robert Israel, Oct 22 2017
  • Mathematica
    smax = 100000; (* upper limit for last term *)
m0 = smax^(1/3) // Ceiling;
f[m_] := f[m] = Module[{c, s, d}, Table[c = CoprimeQ[i^3, j^3]; {s = i^3 + j^3; If[0 < s <= smax && c, s, Nothing], d = j^3 - i^3; If[0 < d <= smax && c, d, Nothing]}, {i, 0, m}, {j, i, m}] // Flatten // Union];
f[m = m0];
f[m += m0];
While[f[m] != f[m - m0], m += m0];
f[m] (* Jean-François Alcover, Jun 28 2023 *)
  • PARI
    upto(lim) = {my(res = List([2]), c, i, j); for(i=1,sqrtnint(lim, 3), for(j=0, sqrtnint(lim - i^3, 3), if(gcd(i, j) == 1, listput(res, c)))); for(i=1, sqrtint(lim\3)+1, for(j = 1, i, if(gcd(i, j) == 1, c = i^3 - (i-j)^3; if(c<=lim, listput(res, c), next(2))))); listsort(res, 1); res} \\ David A. Corneth, Oct 20 2017

A293650 Sum of two (possibly negative) cubes in at least 3 ways (primitive solutions).

Original entry on oeis.org

728, 3367, 4104, 5859, 46683, 65728, 68913, 101528, 124488, 134379, 152551, 155736, 165464, 168112, 184464, 195841, 205352, 289224, 333944, 342657, 402597, 439101, 622232, 625177, 684019, 754299, 757701, 842751, 845208, 1009736, 1016496, 1062936, 1073375
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Primitive means that the 6 summands are coprime. Not every term is the sum of two coprime cubes. a(1) = A047696(3).

Examples

			4104 = 18^3 - 12^3 = 16^3 + 2^3 = 15^3 + 9^3, and 18, -12, 16, 2, 15, 9 are coprime, so 4104 is in the sequence.

Links

Crossrefs

Cf. A051383 (all solutions); A003825 (positive cubes); A293651 (only coprime); A293645, A293647
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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