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 1-4 of 4 results.

A175366 Partial sums of A175365.

Original entry on oeis.org

1, 7, 19, 27, 27, 27, 27, 27, 33, 57, 81, 81, 81, 81, 81, 81, 93, 117, 117, 117, 117, 117, 117, 117, 125, 125, 125, 131, 155, 179, 179, 179, 179, 179, 179, 203, 251, 251, 251, 251, 251, 251, 251, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 275, 287, 311
Offset: 0

Views

Author

R. J. Mathar, Apr 24 2010

Keywords

Comments

Number of integer triples (x,y,z) satisfying |x|^3+|y|^3+|z|^3 <= n, -n <= x,y,z <= n. A variant of A117609 with cubes instead of squares.

A175368 Number of integer 4-tuples (x,y,z,u) satisfying |x|^3 + |y|^3 + |z|^3 + |u|^3 = n, -n <= x,y,z,u <= n.

Original entry on oeis.org

1, 8, 24, 32, 16, 0, 0, 0, 8, 48, 96, 64, 0, 0, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 32, 64, 0, 8, 48, 96, 64, 0, 16, 0, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 192, 0, 0, 0, 0, 0, 0, 64, 0, 0, 24, 96, 96, 0, 0, 0, 0, 0, 96, 192, 8, 48, 96, 64, 0, 0, 96, 0, 48, 192, 192, 0, 0, 0, 0, 0, 96, 224, 64, 0
Offset: 0

Views

Author

R. J. Mathar, Apr 24 2010

Keywords

Comments

A variant of A000118 with cubes instead of squares.

Examples

			a(1) = 8 counts (x,y,z,u) = (-1,0,0,0), (0,-1,0,0), (0,0,-1,0), (0,0,0,-1) and 4 more tuples with -1 replaced by +1.
a(2) = 24 counts (x,y,z,u) = (-1,-1,0,0), (-1,0,-1,0), (-1,0,0,-1), (-1,0,0,1) etc, all variants where two of the 4 values are zero and the other two +1 or -1.

Links

Crossrefs

Cf. A000118 , A175362, A175365.

Programs

  • PARI
    a(n, k=4) = if(n==0, return(1)); if(k <= 0, return(0)); if(k == 1, return(ispower(n,3))); my(count = 0); for(v = 0, sqrtnint(n, 3), count += (2 - (v == 0))*if(k > 2, a(n - v^3, k-1), if(ispower(n - v^3, 3), 2 - (n - v^3 == 0), 0))); count; \\ Daniel Suteu, Aug 15 2021

Formula

Conjectured g.f.: (1 + 2*Sum_{j>=1} x^(j^3))^4.
a(n) = A175365(n) + 2*Sum_{k=1..floor(n^(1/3))} A175365(n - k^3). - Daniel Suteu, Aug 15 2021

A338932 Numbers k such that the Diophantine equation x^3 + y^3 + z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

1, 2, 128, 729, 1458, 4096, 65536, 93312, 2985984, 3906250, 16777216, 28697814, 33554432, 47775744, 80707214, 244140625, 250000000, 387420489, 1836660096, 2847656250, 4715895382, 5165261696, 12230590464, 13841287201, 17179869184, 21208998746, 24461180928
Offset: 1

Views

Author

XU Pingya, Nov 16 2020

Keywords

Comments

The data are derived from the following formula:
(a^3 - 6*t^3)^3 + (a^3 + 6*t^3)^3 + (-6*a*t^2)^3 = 2*a^9;
(4*a^3 - 3*t^3)^3 + (4*a^3 + 3*t^3)^3 + (-6*a*t^2)^3 = 128*a^9 = 2*4^3*a^9;
(9*a^3 - 2*t^3)^3 + (9*a^3 + 2*t^3)^3 + (-6*a*t^2)^3 = 1458*a^9 = 2*9^3*a^9;
(36*a^3 - t^3)^3 + (36*a^3 + t^3)^3 + (-6*a*t^2)^3 = 93312*a^9 = 2*36^3*a^9;
((3*a^3)*t - 9*t^4)^3 + (9*t^4)^3 + (a^4 - 9*a*t^3)^3 = a^12;
((9*a^3)*t - t^4)^3 + (t^4)^3 + (9*a^4 - 3*a*t^3)^3 = 729*a^12 = 9^3*a^12.

Examples

			128 is a term, because (4 - 3*(2*n - 1)^3, 4 + 3*(2*n - 1)^3, -3*(2*n - 1)^2) is a nontrivial primitive parametric solution of x^3 + y^3 + z^3 = 128.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338933.

Programs

  • Mathematica
    t1 = 2*{1, 5, 7, 11, 13}^9;
t2 = 128*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*{1, 3, 5, 7, 9}^9;
t4 = 93312*{1, 2, 3, 4, 5}^9;
t5 = {1, 2, 4, 5, 7}^12;
t6 = 729*{1, 2, 3, 4, 5}^12;
Take[Union[t1, t2, t3, t4, t5, t6], 27]

A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.

Original entry on oeis.org

2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808
Offset: 1

Views

Author

XU Pingya, Nov 16 2020

Keywords

Comments

The data are derived from the following formula:
(a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
(a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
(9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;
(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;
(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.

Examples

			16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.

References

  • R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Crossrefs

Cf. A113490, A173515, A175365, A224215, A226903, A281224, A327586, A338932.

Programs

  • Mathematica
    t1 = 2*Range[23]^6;
t2 = 2*{1, 2, 4, 5, 7, 8}^9;
t3 = 1458*Range[4]^9;
t4 = 2*{1, 5}^12;
t5 = 16*{1, 2, 4}^12;
t6 = 1458*{1, 3}^12;
t7 = 11664*{1, 2, 3}^12;
Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]

Extensions

Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022
Showing 1-4 of 4 results.

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