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.

A239395 Twice prime nonnegative Hurwitz quaternions shown as 4-vectors sorted by norm and then (1,i,j,k) components.

Original entry on oeis.org

2, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 2, 0, 2, 0, 0, 2, 2, 3, 1, 1, 1, 2, 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 0, 2, 2, 2, 4, 2, 0, 0, 4, 0, 2, 0, 4, 0, 0, 2, 3, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 2, 4, 0, 0, 2, 0, 4, 0
Offset: 1

Views

Author

T. D. Noe, Mar 21 2014

Keywords

Comments

The vectors are multiplied by 2 because a Hurwitz quaternion can have half-integer 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.

Links

Crossrefs

Cf. A239393 (Lipschitz quaternions).
Cf. A239396 (number of Hurwitz quaternions having norm prime(n)).

Programs

  • Mathematica
    (* first << Quaternions` *) mx = 5; 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] &]; 2*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]

A240068 Number of prime Lipschitz quaternions having norm prime(n).

Original entry on oeis.org

24, 32, 48, 64, 96, 112, 144, 160, 192, 240, 256, 304, 336, 352, 384, 432, 480, 496, 544, 576, 592, 640, 672, 720, 784, 816, 832, 864, 880, 912, 1024, 1056, 1104, 1120, 1200, 1216, 1264, 1312, 1344, 1392, 1440, 1456, 1536, 1552, 1584, 1600, 1696, 1792
Offset: 1

Views

Author

T. D. Noe, Apr 01 2014

Keywords

Comments

This sequence counts all prime Lipschitz quaternions having a given norm; A239394 counts only the prime nonnegative Lipschitz quaternions.

Links

Crossrefs

Cf. A239393 (prime Lipschitz quaternions), A239394.
Cf. A055669 (number of prime Hurwitz quaternions of norm prime(n)).

Programs

  • Mathematica
    (* first << Quaternions` *)
mx = 17; lst = Flatten[Table[{a, b, c, d}, {a, -mx, mx}, {b, -mx, mx}, {c, -mx, mx}, {d, -mx, 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]

Formula

a(n) = 8 * (prime(n) + 1) = 8 * A008864(n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.