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 1-5 of 5 results.

A239393 Nonnegative prime Lipschitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 1, 1, 2, 0, 0, 1, 0, 2, 0, 1, 0, 0, 2, 0, 2, 1, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 1, 2
Offset: 1

Views

Author

T. D. Noe, Mar 21 2014

Keywords

Comments

A Lipschitz quaternion has all integer components. The norms of quaternions are (rational) primes 2, 3, 5, 7, 11,... A quaternion is commonly written a + b*i + c*j + d*k, where 1, i, j, and k are units.

Examples

			The first six nonnegative prime Lipschitz quaternions are 1+i, 1+j, 1+k, i+j, i+k, and j+k.

Links

Crossrefs

Cf. A239394 (number of Lipschitz quaternions having norm prime(n)).
Cf. A239395 (Hurwitz quaternions).

Programs

  • Mathematica
    (* first << Quaternions` *) mx = 5; lst = Flatten[Table[{a, b, c, d}, {a, 0, mx}, {b, 0, mx}, {c, 0, mx}, {d, 0, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; Sort[q, Norm[#1] < Norm[#2] &]

A239394 Number of prime nonnegative Lipschitz quaternions having norm prime(n).

Original entry on oeis.org

6, 4, 12, 4, 12, 16, 24, 16, 12, 36, 16, 28, 48, 28, 24, 48, 48, 52, 40, 36, 52, 40, 60, 84, 64, 96, 52, 72, 76, 84, 64, 96, 96, 88, 120, 76, 100, 88, 84, 132, 120, 124, 96, 112, 132, 100, 124, 112, 144, 148, 156, 120, 160, 168, 180, 132, 204, 136, 160, 204
Offset: 1

Views

Author

T. D. Noe, Mar 21 2014

Keywords

Comments

For n > 1, there are prime(n) + 1 more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions. - T. D. Noe, Mar 31 2014

Examples

			The six prime nonnegative Lipschitz quaternions having norm 2 are 1+i, 1+j, 1+k, i+j, i+k, and j+k.

Links

Crossrefs

Cf. A239393 (prime Lipschitz quaternions).
Cf. A239395 (prime Hurwitz quaternions).

Programs

  • Mathematica
    (* first << Quaternions` *) mx = 17; lst = Flatten[Table[{a, b, c, d}, {a, 0, mx}, {b, 0, mx}, {c, 0, mx}, {d, 0, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; q2 = Sort[q, Norm[#1] < Norm[#2] &]; Transpose[Tally[(Norm /@ q2)^2]][[2]]

A239396 Number of prime nonnegative Hurwitz quaternions having norm prime(n).

Original entry on oeis.org

6, 8, 18, 12, 24, 30, 42, 36, 36, 66, 48, 66, 90, 72, 72, 102, 108, 114, 108, 108, 126, 120, 144, 174, 162, 198, 156, 180, 186, 198, 192, 228, 234, 228, 270, 228, 258, 252, 252, 306, 300, 306, 288, 306, 330, 300, 336, 336, 372, 378, 390, 360, 402, 420, 438
Offset: 1

Views

Author

T. D. Noe, Mar 21 2014

Keywords

Comments

For n > 1, there are prime(n) + 1 more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions. - T. D. Noe, Mar 31 2014

Examples

			The six prime nonnegative Hurwitz quaternions having norm 2 are 1+i, 1+j, 1+k, i+j, i+k, and j+k.

Links

Crossrefs

Cf. A239393 (prime Lipschitz quaternions).
Cf. A239395 (prime Hurwitz quaternions).

Programs

  • Mathematica
    (* first << Quaternions` *) mx = 17; lst = Flatten[Table[{a, b, c, d}/2, {a, 0, mx}, {b, 0, mx}, {c, 0, mx}, {d, 0, mx}], 3]; q = Select[lst, Norm[Quaternion @@ #] < mx^2 && PrimeQ[Quaternion @@ #, Quaternions -> True] &]; q2 = Sort[q, Norm[#1] < Norm[#2] &]; Take[Transpose[Tally[(Norm /@ q2)^2]][[2]], mx]

A240066 Twice prime nonnegative octonion integers shown as 8-vectors sorted by norm and then real and 7 imaginary components.

Original entry on oeis.org

2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 1, 0, 1, 2, 1, 1, 0, 0, 0, 1, 1
Offset: 1

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

The norm of an octonion is the sum of the squares of its components. Some authors use the square root of that number. There are 309 nonnegative octonions having norm 2. The first 11 are shown above. Counting octonions having both positive and negative entries, there are 9328 having norm 2; see A240067.

Examples

			The first primes listed are 1 + i and 1 + (i+j+k+l)/2.

References

  • John H. Conway and Derek A. Smith, On Quaternions and Octonions, CRC, 2003.

Links

Crossrefs

Cf. A239395 (twice prime nonnegative Hurwitz quaternions), A240067.

Programs

  • Mathematica
    Reverse[Rest[Union[Flatten[Table[If[(a/2)^2 + (b/2)^2 + (c/2)^2 + (d/2)^2 + (e/2)^2 + (f/2)^2 + (g/2)^2 + (h/2)^2 == 2, {a, b, c, d, e, f, g, h}, {0}], {a, 0, 2}, {b, 0, 2}, {c, 0, 2}, {d, 0, 2}, {e, 0, 2}, {f, 0, 2}, {g, 0, 2}, {h, 0, 2}], 7]]]]

A240067 Twice prime octonion integers shown as 8-vectors sorted by norm and then real and 7 imaginary components.

Original entry on oeis.org

2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 1, 1, 1, 0, 0, 0, 2, 1, 1, 1, 0, 1, 0, 0, 2, 1, 1, 1, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, -1, 2, 1, 1, 1, 0, 0, -1, 0, 2, 1, 1, 1, 0, -1, 0, 0, 2, 1, 1, 1, -1, 0, 0, 0, 2, 1, 1, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 1, 0
Offset: 1

Views

Author

T. D. Noe, Mar 31 2014

Keywords

Comments

The norm of an octonion is the sum of the squares of its components. Some authors use the square root of that number. There are 9328 octonions having norm 2. The first 11 are shown above. Counting octonions having just nonnegative entries, there are 309 having norm 2; see A240066.

References

  • John H. Conway and Derek A. Smith, On Quaternions and Octonions, CRC, 2003.

Links

Crossrefs

Cf. A239395 (twice prime nonnegative Hurwitz quaternions), A240066.

Programs

  • Mathematica
    Reverse[Rest[Union[Flatten[Table[If[(a/2)^2 + (b/2)^2 + (c/2)^2 + (d/2)^2 + (e/2)^2 + (f/2)^2 + (g/2)^2 + (h/2)^2 == 2, {a, b, c, d, e, f, g, h}, {0}], {a, -2, 2}, {b, -2, 2}, {c, -2, 2}, {d, -2, 2}, {e, -2, 2}, {f, -2, 2}, {g, -2, 2}, {h, -2, 2}], 7]]]]
Showing 1-5 of 5 results.

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