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.

User: Rosalie Fay

Rosalie Fay's wiki page.

Rosalie Fay has authored 7 sequences.

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

A293649 Sum of two coprime positive cubes in at least 2 ways, but not the sum of 2 non-coprime positive cubes.

Original entry on oeis.org

1729, 20683, 40033, 149389, 195841, 327763, 443889, 684019, 704977, 1845649, 2048391, 2418271, 2691451, 3242197, 3375001, 4342914, 4931101, 5318677, 5772403, 5799339, 6058747, 7620661, 8872487, 9443761, 10702783, 10765603, 13623913, 14916727
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

			14916727 = 246^3 + 31^3 = 240^3 + 103^3 and 246 & 31 are coprime, as are 240 & 103, but it is not also the sum of cubes of 2 non-coprime positive integers, so 14916727 is in the sequence.

Links

Crossrefs

Cf. A023050 (3 ways); A272885 (cubefree with positive cubes).
Cf. A159843, A293648 (allows negatives).

Programs

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

A293651 Sum of two (possibly negative) coprime cubes in at least 3 ways, but not the sum of 2 noncoprime cubes.

Original entry on oeis.org

3367, 68913, 152551, 195841, 625177, 684019, 1147627, 1548729, 2113921, 2628073, 2985983, 3242197, 3442887, 4488211, 4663295, 4931101, 5318677, 7194889, 8741691, 9667693, 14110579, 15072967, 15438185, 16776487, 21463407, 22910797, 24769502, 26122131, 26460217
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

			21463407 = 271^3 + 116^3 = 284^3 - 113^3 = 368^3 - 305^3, and 271 & 116 are coprime, etc., so 21463407 is in the sequence.

Links

Crossrefs

Cf. A023050 (positive cubes); A293650 (allows noncoprime); A293646, A293648

A293646 Sum of two (possibly negative) coprime cubes, but not the sum of 2 non-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, 169, 215, 217, 218, 271, 279, 316, 331, 335, 341, 342, 344, 351, 370, 386, 387, 397, 407, 468, 469, 485, 511, 539, 547, 559, 602, 604, 631, 637, 657, 665, 721, 730, 737, 793, 817, 819
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

			344 = 7^3 + 1^3 and 344 is not also the sum of cubes of 2 non-coprime integers, so 344 is in the sequence.
152 = 6^3 + (-4)^3 and 6 and -4 are not coprime, so 152 is not in the sequence.

Links

Crossrefs

Cf. A020895 (cubefree); A293645 (allows non-coprime); A293648, A293651

Programs

  • Mathematica
    s[n_] := CoprimeQ @@@ ({x, y} /. Solve[n == x^3 + y^3, {x, y}, Integers]);
Reap[For[k = 1, k < 2000, k++, If[Union[s[k]] == {True}, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Feb 02 2023 *)

A293647 Positive numbers that are the sum of two (possibly negative) cubes in at least 2 ways (primitive solutions).

Original entry on oeis.org

91, 152, 189, 217, 513, 721, 728, 999, 1027, 1729, 3087, 3367, 4104, 4706, 4921, 4977, 5256, 5859, 6832, 7657, 8587, 8911, 9919, 10621, 10712, 12663, 12691, 12824, 14911, 15093, 15561, 16120, 16263, 20683, 21014, 23058, 23877, 25669, 27937, 28063, 31519, 32984
Offset: 1

Views

Author

Rosalie Fay, Oct 16 2017

Keywords

Comments

Primitive means that the 4 summands are coprime.
Not every term is the sum of two coprime cubes.
a(1) = A047696(2).

Examples

			189 = 4^3 + 5^3 = 6^3 + (-3)^3 and 4, 5, 6, -3 are coprime, so 189 is in the sequence.
35208 = 34^3 + (-16)^3 = 33^3 + (-9)^3 and 34, -16, 33, -9 are coprime, so 35208 is in the sequence.

Links

Crossrefs

Cf. A051347 (all solutions); A018850 (positive cubes); A293648 (only coprime); A293645, A293650

Programs

  • Maple
    g:= proc(s,n) local x;
  x:= s/2 + sqrt(12*n/s-3*s^2)/6;
  if not x::integer then return NULL fi;
  [x,s - x];
end proc:
filter:= proc(n)
  local pairs, i,j;
  pairs:= map(g, numtheory:-divisors(n),n);
  for i from 2 to nops(pairs) do
    for j from 1 to i-1 do
      if igcd(op(pairs[i]),op(pairs[j]))=1 then return true fi
  od od;
  false
end proc:
select(filter, [seq(seq(9*i+j,j=[1,2,7,8,9]),i=0..4000)]); # Robert Israel, Oct 22 2017
  • Mathematica
    g[s_, n_] := Module[{x}, x = s/2 + Sqrt[12*n/s - 3*s^2]/6;   If[!IntegerQ[x], Return[Nothing]]; {x, s - x}];
filter[n_] := Module[{pairs, i, j}, pairs = g[#, n]& /@ Divisors[n]; For[i = 2, i <= Length[pairs], i++,For[j = 1, j <= i - 1, j++, If[GCD @@ Join[pairs[[i]], pairs[[j]]] == 1, Return[True]]]]; False];
Select[Flatten[Table[Table[9*i + j, {j, {1, 2, 7, 8, 9}}], {i, 0, 4000}]], filter] (* Jean-François Alcover, May 28 2023, after Robert Israel *)

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

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