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.

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A336205 Numbers k that can be expressed as x^3 + y^3 + z^3 with x^2 + y^2 + z^2 <= k where x, y, z are integers.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 8, 9, 10, 15, 16, 17, 18, 19, 20, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 43, 45, 46, 48, 53, 54, 55, 56, 57, 60, 61, 62, 63, 64, 65, 66, 69, 71, 72, 73, 80, 81, 83, 88, 90, 91, 92, 97, 98, 99, 100, 101, 106, 109, 116, 117, 118, 119, 120, 123, 124, 125, 126, 127, 128, 129, 132
Offset: 1

Views

Author

Altug Alkan, Jul 12 2020

Keywords

Comments

See A336240 for border case x^2 + y^2 + z^2 = x^3 + y^3 + z^3.
What is the natural density of this sequence?
There are infinitely many infinite parametric families of solutions which have negative values in (x,y,z). For example, 8*(3*a-1)^2*m^6 + 12*(3*a-1)*(a-1)*m^4 - 6*(2*a-1)*m^2 + 2*a^3 + 1 are terms for all a >= 0, m >= 0. (x = 1 - (6*a-2)*m^2, y = a - m*(1-(6*a-2)*m^2), z = a + m*(1-(6*a-2)*m^2)). - Altug Alkan, Jul 17 2020
By definition, corresponding (x,y,z) variables are produced by equation x^3 + y^3 + z^3 = x^2 + y^2 + z^2 + t with t >= 0. That is, x^2*(x-1) + y^2*(y-1) + z^2*(z-1) >= 0. Conjecture: Every even integer can be represented as x^2*(x-1) + y^2*(y-1) + z^2*(z-1) where x, y, z are integers. - Altug Alkan, Jul 19 2020

Examples

			11 is not a term because there is no (x,y,z) with x^2 + y^2 + z^2 <= 11 when x^3 + y^3 + z^3 = 11.
18 is a term because (-1)^3 + (-2)^3 + 3^3 = 18 and (-1)^2 + (-2)^2 + 3^2 <= 18.
61 is a term because (-4)^3 + 0^3 + 5^3 = 61 and (-4)^2 + 0^2 + 5^2 <= 61.
354 is a term because (-11)^3 + (-8)^3 + 13^3 = (-11)^2 + (-8)^2 + 13^2 = 354.

Links

Crossrefs

Cf. A004825 (subsequence), A060464 (supersequence), A336240.

Programs

  • C
    See Links section.
  • Maple
    filter:= proc(n) local x,y,z,e1,e2;
  for x from 0 while 3*x^2 <= n do
    for y from 0 while x^2 + 2*y^2 <= n do
      for e1 in [-1,1] do for e2 in [-1,1] do
        z:= surd(n + e1*x^3 + e2*y^3,3);
        if z::integer and x^2 + y^2 + z^2 <= n then return true fi;
  od od od od;
  false
end proc:
select(filter, [$0..200]); # Robert Israel, Jul 12 2020
  • Mathematica
    filter[n_] := Module[{x, y, z, e1, e2},
  For[x = 0, 3*x^2 <= n, x++,
    For[y = 0, x^2 + 2*y^2 <= n, y++,
      For[e1 = -1, e1 <= 1, e1 += 2, For[e2 = -1, e2 <= 1, e2 += 2,
        z = (n + e1*x^3 + e2*y^3)^(1/3);
        If[IntegerQ[z] && x^2 + y^2 + z^2 <= n, Return[True]]
  ]]]]; False];
Select[Range[0, 200], filter] (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)

A265227 Nonnegative m for which k*floor(m^2/9) = floor(k*m^2/9), with 2 < k < 9.

Original entry on oeis.org

0, 1, 3, 6, 8, 9, 10, 12, 15, 17, 18, 19, 21, 24, 26, 27, 28, 30, 33, 35, 36, 37, 39, 42, 44, 45, 46, 48, 51, 53, 54, 55, 57, 60, 62, 63, 64, 66, 69, 71, 72, 73, 75, 78, 80, 81, 82, 84, 87, 89, 90, 91, 93, 96, 98, 99, 100, 102, 105, 107, 108, 109, 111, 114
Offset: 1

Views

Author

Bruno Berselli, Dec 06 2015

Keywords

Comments

Also, nonnegative m congruent to 0, 1, 3, 6 or 8 (mod 9). The product of any two terms belongs to the sequence and so also a(n)^2, a(n)^3, a(n)^4, etc.
Integers x >= 0 satisfying k*floor(x^2/9) = floor(k*x^2/9) with k >= 0:
k = 0, 1: x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... (A001477);
k = 2: x = 0, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 15, ... (A060464);
k = 3..8: x = 0, 1, 3, 6, 8, 9, 10, 12, 15, 17, 18, ... (this sequence);
k > 8: x = 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ... (A008585).
Primes in sequence: 3, 17, 19, 37, 53, 71, 73, 89, 107, 109, 127, ...

Links

Crossrefs

Cf. A001477, A008585, A060464.
Cf. similar sequences listed in A265188.

Programs

  • Magma
    [n: n in [0..120] | 3*Floor(n^2/9) eq Floor(3*n^2/9)]; /* or, by the definition: */ K:=[3..8]; [: k in K];
  • Mathematica
    Select[Range[0, 120], 3 Floor[#^2/9] == Floor[3 #^2/9] &]
Select[Range[0, 120], MemberQ[{0, 1, 3, 6, 8}, Mod[#, 9]] &]
LinearRecurrence[{1, 0, 0, 0, 1, -1}, {0, 1, 3, 6, 8, 9}, 70]
  • Sage
    [n for n in (0..120) if 3*floor(n^2/9) == floor(3*n^2/9)]

Formula

G.f.: x^2*(1 + 2*x + 3*x^2 + 2*x^3 + x^4)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n>6.

A307585 Positive sums of two distinct cubes (of arbitrary sign).

Original entry on oeis.org

1, 7, 8, 9, 19, 26, 27, 28, 35, 37, 56, 61, 63, 64, 65, 72, 91, 98, 117, 124, 125, 126, 127, 133, 152, 169, 189, 208, 215, 216, 217, 218, 224, 243, 271, 279, 280, 296, 316, 331, 335, 341, 342, 343, 344, 351, 370, 386, 387, 397, 407, 448, 468, 469, 485, 488, 504, 511, 512, 513, 520, 539, 547, 559
Offset: 1

Views

Author

Robert Israel, Apr 15 2019

Keywords

Comments

All terms == 0, 1, 2, 7 or 8 (mod 9).

Examples

			a(3) = 8 = 0^3 + 2^3.
a(4) = 9 = 1^3 + 2^3.
a(5) = 19 = (-2)^3 + 3^3.

Links

Crossrefs

Contained in A045980. Contains A024670.
Primes in this sequence: A002407.
Cf. A060464.

Programs

  • Maple
    filter:= proc(n) local d, dp, r;
   for d in numtheory:-divisors(n) do
     dp:= n/d;
     r:= 12*dp - 3*d^2;
     if r > 0 and issqr(r) and (sqrt(r)/6 + d/2)::integer then return true fi
   od;
   false
end proc:
select(filter, [$0..1000]);
  • Mathematica
    filterQ[n_] := Module[{d, dp, r}, Catch[Do[dp = n/d; r = 12 dp - 3 d^2; If[r > 0 && IntegerQ[Sqrt[r]] && IntegerQ[Sqrt[r]/6 + d/2], Throw[True]], {d, Divisors[n]}]; False]];
Select[Range[1000], filterQ] (* Jean-François Alcover, Oct 17 2020, after Maple *)
Previous Showing 11-13 of 13 results.

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