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A055669 -id:A055669 - OEIS
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Search: a055669 -id:a055669
Displaying 1-5 of 5 results found. page 1
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A055670 a(n) = prime(n) - (-1)^prime(n). +10
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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
REFERENCES
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.
LINKS
Shawn A. Broyles, Table of n, a(n) for n = 1..20000
FORMULA
a(n) = prime(n)+1 = A008864(n) for n >= 2. a(n) = A055669(n)/24.
EXAMPLE
a(1) = 2 - (-1)^2 = 1, a(2) = 3 - (-1)^3 = 4.
MATHEMATICA
Join[{1}, Prime[Range[2, 70]]+1] (* Harvey P. Dale, Oct 29 2013 *)
CROSSREFS
Cf. A000040, A006093.
Cf. A055669-A055672.
a(n) = A083503(p) for n>1.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
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
STATUS
approved
A055673 Absolute values of norms of primes in ring of integers Z[sqrt(2)]. +10
11
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
REFERENCES
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VII.
LINKS
T. D. Noe, Table of n, a(n) for n = 1..1000
FORMULA
Consists of 2; rational primes = +-1 (mod 8); and squares of rational primes = +-3 (mod 8).
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
Cf. A055026-A055029, A055669-A055672, A118916, A118917.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
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
STATUS
approved
A055672 Number of right-inequivalent prime Hurwitz quaternions of norm n. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Two primes are considered right-equivalent if they differ by right multiplication by one of the 24 units.
REFERENCES
L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = A055671(n)/24.
MATHEMATICA
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 *)
CROSSREFS
Cf. A055669-A055671.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
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.
STATUS
approved
A055671 Number of prime Hurwitz quaternions of norm n. +10
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
L. E. Dickson, Algebras and Their Arithmetics, Dover, 1960, Section 91.
LINKS
R. J. Mathar, Table of n, a(n) for n = 0..10000
FORMULA
a(n) = number of vectors of norm n in D_4 lattice (A004011) if n is a prime, otherwise a(n) = 0.
MATHEMATICA
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 *)
CROSSREFS
Cf. A055669-A055672.
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Jun 09 2000
EXTENSIONS
More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Jun 10 2005
STATUS
approved
A240068 Number of prime Lipschitz quaternions having norm prime(n). +10
1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
This sequence counts all prime Lipschitz quaternions having a given norm; A239394 counts only the prime nonnegative Lipschitz quaternions.
LINKS
Table of n, a(n) for n=1..48.
Wikipedia, Hurwitz quaternion
FORMULA
a(n) = 8 * (prime(n) + 1) = 8 * A008864(n).
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]
CROSSREFS
Cf. A239393 (prime Lipschitz quaternions), A239394.
Cf. A055669 (number of prime Hurwitz quaternions of norm prime(n)).
KEYWORD
nonn
AUTHOR
T. D. Noe, Apr 01 2014
STATUS
approved
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Last modified September 7 13:22 EDT 2024. Contains 375730 sequences. (Running on oeis4.)