A055669 -id:A055669 - cp's OEIS frontend

A055670 a(n) = prime(n) - (-1)^prime(n).

1, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

N. J. A. Sloane, Jun 09 2000

Number of right-inequivalent prime Hurwitz quaternions of norm p, where p = n-th rational prime (indexed by A000040).
Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units. - N. J. A. Sloane
Start of n-th run of consecutive nonprime numbers. Since 2 is the only even prime, for all other prime numbers the expression "- (-1)^(n-th prime)" works out to "+ 1." - Alonso del Arte, Oct 18 2011

			a(1) = 2 - (-1)^2 = 1, a(2) = 3 - (-1)^3 = 4.

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.
Lynn Arthur Steen and J. Arthur Seebach, Jr., Counterexamples in Topology, Dover, New York, 1978, page 134.

Shawn A. Broyles, Table of n, a(n) for n = 1..20000

Cf. A000040, A006093.
Cf. A055669-A055672.
a(n) = A083503(p) for n>1.

Join[{1},Prime[Range[2,70]]+1] (* Harvey P. Dale, Oct 29 2013 *)

a(n) = prime(n)+1 = A008864(n) for n >= 2. a(n) = A055669(n)/24.

More terms from David W. Wilson, May 02 2001
I would also like to get the sequences of inequivalent prime Hurwitz quaternions, where two primes are considered equivalent if they differ by left or right multiplication by one of the 24 units. This will give two more sequences, analogs of A055670 and A055672.
Edited by N. J. A. Sloane, Aug 16 2009

A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)].

Original entry on oeis.org

2, 7, 9, 17, 23, 25, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 121, 127, 137, 151, 167, 169, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 361, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593

Offset: 1

N. J. A. Sloane, Jun 09 2000

The integers have the form z = a + b*sqrt(2), a and b rational integers. The norm of z is a^2 - 2*b^2, which may be negative.

L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VII.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A055026-A055029, A055669-A055672, A118916, A118917.

maxNorm = 593; s1 = Select[Range[-1, maxNorm, 8], PrimeQ]; s2 = Select[Range[1, maxNorm, 8], PrimeQ]; s3 = Select[Range[-3, Sqrt[maxNorm], 8], PrimeQ]^2; s4 = Select[Range[3, Sqrt[maxNorm], 8], PrimeQ]^2; Union[{2}, s1, s2, s3, s4] (* Jean-François Alcover, Dec 07 2012, from formula *)

is(n)=!!if(isprime(n), setsearch([1,2,7],n%8), issquare(n,&n) && isprime(n) && setsearch([3,5], n%8)) \\ Charles R Greathouse IV, Sep 10 2016

Consists of 2; rational primes = +-1 (mod 8); and squares of rational primes = +-3 (mod 8).

I would also like to get the sequences (analogous to A055027 and A055029) giving the number of inequivalent primes mod units. Of course now there are infinitely many units.

More terms from Franklin T. Adams-Watters, May 05 2006

A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.

Original entry on oeis.org

0, 0, 1, 4, 0, 6, 0, 8, 0, 0, 0, 12, 0, 14, 0, 0, 0, 18, 0, 20, 0, 0, 0, 24, 0, 0, 0, 0, 0, 30, 0, 32, 0, 0, 0, 0, 0, 38, 0, 0, 0, 42, 0, 44, 0, 0, 0, 48, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 60, 0, 62, 0, 0, 0, 0, 0, 68, 0, 0, 0, 72, 0, 74, 0, 0, 0, 0, 0, 80, 0, 0, 0, 84, 0, 0, 0, 0, 0, 90

Offset: 0

N. J. A. Sloane, Jun 09 2000

Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units.

The extension desired below does not exist in the following sense: Let q1 ~R q2 be the equivalence defined by q1 = q2*u (i.e. if q1 and q2 any two HQ's, a unit u exists that solves this). Let q1 ~L q2 be the equivalence defined by q1 = u*q2 (i.e. if q1 and q2 any two HQ's a unit u exists that solves this.) If we define a relation ~RL such that q1 ~RL q2 means (q1 ~R q2 or q1 ~L q2), this relation is not transitive, i.e., not an equivalence. Cause: q1 = q2*u1, q2 = u2*q3, i.e., q1 ~R q2, q2 ~L q3 does not always have a solution with either q1 = q3*u3 or q1= u3*q3. There are pairs of u1 and u2 out of the 24*24 cases where q1 ~L q3 or q1 ~L q3 cannot be solved with any u3. - R. J. Mathar, Aug 05 2025

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

R. J. Mathar, Table of n, a(n) for n = 0..10000
L. E. Dickson, Algebras and Their Arithmetics, U. Chicago Press, 1923, Section 91.

Cf. A055669 A055670 (zeros removed), A055671, A385603 (equivalence up to left-and-right multiplication).

A055671[n_] := If[PrimeQ[n], Reduce[a^2 + b^2 + c^2 + d^2 == 4n, {a, b, c, d}, Integers] // Length, 0]; a[n_] := A055671[n]/24; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 22 2016 *)

a(n) = A055671(n)/24.

a(n) = A000593(n) * A010051(n). - R. J. Mathar, Aug 02 2025

I would also like to get the sequences of inequivalent prime Hurwitz quaternions, where two primes are considered equivalent if they differ by left or right multiplication by one of the 24 units. This will give two more sequences, analogs of A055670 and A055672.

A055671 Number of prime Hurwitz quaternions of norm n.

Original entry on oeis.org

0, 0, 24, 96, 0, 144, 0, 192, 0, 0, 0, 288, 0, 336, 0, 0, 0, 432, 0, 480, 0, 0, 0, 576, 0, 0, 0, 0, 0, 720, 0, 768, 0, 0, 0, 0, 0, 912, 0, 0, 0, 1008, 0, 1056, 0, 0, 0, 1152, 0, 0, 0, 0, 0, 1296, 0, 0, 0, 0, 0, 1440, 0, 1488, 0, 0, 0, 0, 0, 1632, 0, 0, 0, 1728, 0, 1776, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 09 2000

L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.

R. J. Mathar, Table of n, a(n) for n = 0..10000

Cf. A055669 (zeros removed), A055670, A055671, A055672.

a[p_?PrimeQ] := Reduce[ a^2 + b^2 + c^2 + d^2 == 4p, {a, b, c, d}, Integers] // Length; a[] = 0; Table[ a[n], {n, 0, 78}] (* _Jean-François Alcover, Oct 03 2012 *)

a(n) = number of vectors of norm n in D_4 lattice (A004011) if n is a prime, otherwise a(n) = 0.

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 10 2005

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

T. D. Noe, Apr 01 2014

nonn

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

Wikipedia, Hurwitz quaternion

Cf. A239393 (prime Lipschitz quaternions), A239394.

Cf. A055669 (number of prime Hurwitz quaternions of norm prime(n)).

(* 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]

a(n) = 8 * (prime(n) + 1) = 8 * A008864(n).

cp's OEIS Frontend

A055670 a(n) = prime(n) - (-1)^prime(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)].

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A055671 Number of prime Hurwitz quaternions of norm n.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula