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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A334736 Dimensions d such that the integer lattice Z^d does not contain the vertices of a regular d-simplex.

Original entry on oeis.org

2, 4, 5, 6, 10, 12, 13, 14, 16, 18, 20, 21, 22, 26, 28, 29, 30, 32, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 50, 52, 53, 54, 56, 58, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 82, 84, 85, 86, 88, 90, 92, 93, 94, 96, 98, 100, 101, 102, 104, 106, 108
Offset: 1

Views

Author

Harry Richman, May 08 2020

Keywords

Comments

List contains d such that (1) d is even and d+1 is not a square, or (2) d == 1 (mod 4) and d+1 is not a sum of two squares; proved by Schoenberg.

Examples

			2 is in the list because there is no equilateral triangle in the plane whose vertices all have integer coordinates.
3 is not in the list because there is a regular tetrahedron in space whose vertices have integer coordinates; e.g. (1,1,0), (1,0,1), (0,1,1), (0,0,0).

Links

Crossrefs

Complement of A096315.
Cf. A000290, A001481, A022544, A097269.

Extensions

More terms from Jinyuan Wang, May 09 2020

A339273 Sums of two nonzero even squares.

Original entry on oeis.org

8, 20, 32, 40, 52, 68, 72, 80, 100, 104, 116, 128, 136, 148, 160, 164, 180, 200, 208, 212, 232, 244, 260, 272, 288, 292, 296, 320, 328, 340, 356, 360, 388, 392, 400, 404, 416, 424, 436, 452, 464, 468, 488, 500, 512, 520, 544, 548, 580, 584, 592, 596, 612, 628, 640, 648, 656
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 24 2020

Keywords

Examples

			20 is in the sequence since it is the sum of two nonzero even squares, 2^2 + 4^2 = 4 + 16 = 20.

Crossrefs

Cf. A000404, A010052, A057653, A097269.

Programs

  • Mathematica
    Table[If[Sum[Mod[i + 1, 2] Mod[n - i + 1, 2] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 700}] // Flatten

Formula

a(n) = 4*A000404(n).
Characteristic function: sign(Sum_{k=1..floor(n/2)} ((k+1) mod 2) * ((n-k+1) mod 2) * c(k) * c(n-k)), where c is the characteristic function of squares (A010052).

A339955 Numbers that are the sum of an odd square s and an even square t such that 0 < s < t.

Original entry on oeis.org

5, 17, 25, 37, 45, 61, 65, 73, 89, 101, 109, 113, 125, 145, 149, 153, 169, 181, 193, 197, 205, 221, 225, 245, 257, 265, 277, 281, 305, 317, 325, 333, 337, 349, 365, 373, 377, 401, 405, 409, 425, 445, 449, 481, 485, 493, 509, 521, 533, 549, 565, 569, 577, 585, 601, 605, 613
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 24 2020

Keywords

Examples

			17 is in the sequence since 1^2 + 4^2 = 17, 1 is odd, 16 is even, and 0 < 1 < 16.

Crossrefs

Cf. A000404, A010052, A057653, A097269.

Programs

  • Mathematica
    Table[If[Sum[Mod[i, 2] Mod[n - i + 1, 2] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 700}] // Flatten

A339956 Numbers that are the sum of an even square s and an odd square t such that 0 < s < t.

Original entry on oeis.org

13, 29, 41, 53, 65, 85, 97, 117, 125, 137, 145, 157, 173, 185, 205, 221, 229, 233, 241, 261, 269, 289, 293, 305, 313, 325, 353, 365, 369, 377, 389, 397, 421, 425, 433, 445, 457, 461, 477, 485, 505, 533, 541, 545, 557, 565, 585, 593, 617, 629, 637, 641, 661, 673, 685, 689, 697
Offset: 1

Views

Author

Wesley Ivan Hurt, Dec 24 2020

Keywords

Examples

			29 is in the sequence since 2^2 + 5^2 = 29, 4 is even, 25 is odd, and 0 < 4 < 25.

Crossrefs

Cf. A000404, A010052, A057653, A097269.

Programs

  • Mathematica
    Table[If[Sum[Mod[i + 1, 2] Mod[n - i, 2] (Floor[Sqrt[i]] - Floor[Sqrt[i - 1]]) (Floor[Sqrt[n - i]] - Floor[Sqrt[n - i - 1]]), {i, Floor[n/2]}] > 0, n, {}], {n, 700}] // Flatten
Previous Showing 11-14 of 14 results.

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