A010527 -id:A010527 - cp's OEIS frontend

A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

1, 3, 3, 9, 7, 4, 5, 9, 6, 2, 1, 5, 5, 6, 1, 3, 5, 3, 2, 3, 6, 2, 7, 6, 8, 2, 9, 2, 4, 7, 0, 6, 3, 8, 1, 6, 5, 2, 8, 5, 9, 7, 3, 7, 3, 0, 9, 4, 8, 0, 9, 6, 8, 5, 9, 7, 2, 0, 9, 6, 5, 1, 0, 2, 7, 4, 0, 3, 3, 4, 9, 1, 5, 4, 5, 5, 9, 9, 9, 8, 1, 4, 5, 9, 4, 2, 6, 9, 0, 6

Offset: 0

Paolo Xausa, Jul 30 2024

			0.133974596215561353236276829247063816528597373...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Eric Weisstein's World of Mathematics, Sagitta
Index entries for algebraic numbers, degree 2.

Essentially the same as A334843.
Cf. A010527 (apothem), A104956 (area).
Cf. sagitta of other polygons with unit side length: A020769 (triangle), A174968 (square), A375068 (pentagon), A374972 (heptagon), A375070 (octagon), A375153 (9-gon), A375189 (10-gon), A375192 (11-gon), A375194 (12-gon).
Cf. A010527, A019913, A152422, A332133.

First[RealDigits[Tan[Pi/12]/2, 10, 100]]

tan(Pi/12)/2 \\ Charles R Greathouse IV, Feb 04 2025

polrootsreal(4*x^2-8*x+1)[1] \\ Charles R Greathouse IV, Feb 04 2025

Equals tan(Pi/12)/2 = A019913/2.
Equals 1 - sqrt(3)/2 = 1 - A010527.
Equals A152422^2 = (1 - A332133)^2. - Hugo Pfoertner, Jul 30 2024
Equals A334843-1/2. - R. J. Mathar, Aug 02 2024

A003630 Inert rational primes in Q[sqrt(3)].

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 199, 211, 223, 233, 257, 269, 271, 281, 283, 293, 307, 317, 331, 353, 367, 379, 389, 401, 439, 449, 461, 463, 487, 499, 509, 521, 523, 547, 557, 569, 571, 593

Offset: 1

N. J. A. Sloane, Mira Bernstein

Primes p such that p divides 3^(p-1)/2 + 1. - Cino Hilliard, Sep 04 2004
Primes p such that 1 + 4*x + x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011
Conjecture: Primes congruent to {5, 7} mod 12. - Vincenzo Librandi, Aug 06 2012
The above conjecture is correct. In fact, this is the sequence of primes p such that Kronecker(12,p) = -1 (12 is the discriminant of Q[sqrt(3)]), that is, odd primes that have 3 as a quadratic nonresidue. - Jianing Song, Nov 21 2018
Conjecture: Let r(n) = (a(n) - 1)/(a(n) + 1) if a(n) mod 4 = 1, (a(n) + 1)/(a(n) - 1) otherwise; then Product_{n>=1} r(n) = (2/3) * (4/3) * (8/9) * (10/9) * (14/15) * ... = sqrt(3)/2. (See A010527.) We see that the sum of the numerator and denominator of each fraction equals the corresponding term of the sequence: 2 + 3 = 5, 4 + 3 = 7, 8 + 9 = 17, ... - Dimitris Valianatos, Mar 26 2017

			Since (-1)*(1 - sqrt(3))*(1 + sqrt(3)) = 2, 2 is not in the sequence.
3 is not in the sequence for obvious reasons.
x^2 == 3 (mod 5) has no solution, which means that 5 is an inert prime in Z[sqrt(3)]. Therefore, 5 is in the sequence.

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Cf. A010527, A097933.

Select[Prime[Range[2, 200]], JacobiSymbol[3, #] == -1 &] (* Alonso del Arte, Mar 26 2017 *)

{a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( 12, m )== -1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

A066408 Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.

Original entry on oeis.org

2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, 2888387, 4043119

Offset: 1

Mike Oakes, Dec 24 2001

Analog of Mersenne primes in Eisenstein integers.
The norm of a + b * ω is (a + b * ω) * (a + b * ω^2) = a^2 + a*b + b^2.
Indices for which the Eisenstein-Mersenne numbers are primes. The p-th Eisenstein-Mersenne number can be written as 3^p - Legendre(3, p) * 3^((p + 1)/2) + 1. Note the enormous gap between 23743 and 255361. A modified version of Chris Nash's PFGW program was used to find the last term. - Jeroen Doumen (doumen(AT)win.tue.nl), Oct 31 2002
Let q be the integer quaternion (3 + i + j + k)/2. Then q^n - 1 is a quaternion prime for these n; that is, the norm of q^n - 1 is a rational prime. - T. D. Noe, Feb 02 2005
The actual norms also belong to the class of Generalized Unique primes (see Links section), that is primes which have a period of expansion of 1/p (in some general, non-decimal system) that it shares with no other prime. - Serge Batalov, Mar 29 2014
Next term > 4400000. - Serge Batalov, Jun 20 2023

			For n = 7, (1 - ω)^7 - 1 has norm 2269, a prime.
Or, for p = 7, 3^7 + 3^4 + 1 = 2269, which is prime.

P. H. T. Beelen, Algebraic geometry and coding theory, Ph.D. Thesis, Eindhoven, The Netherlands, September 2001.
J. M. Doumen, Ph.D. Thesis, Eindhoven, The Netherlands, to appear.
Mike Oakes, posting to primenumbers(AT)yahoogroups.com, Dec 24 2001.

Pedro Berrizbeitia and Boris Iskra, Gaussian Mersenne and Eisenstein Mersenne primes, Mathematics of Computation 79 (2010), pp. 1779-1791.
Chris Caldwell, The largest known primes
Chris Caldwell, Generalized Unique primes
Mike Oakes, Eisenstein Mersenne and Fermat primes
Mike Oakes, Eisenstein Mersenne and Fermat primes, message 4607 in primenumbers Yahoo group, Dec 24, 2001.
Mike Oakes, A new series of Mersenne-like Gaussian primes
Mike Oakes, Posting to the Number Theory list, Dec 27 2005.
K. Pershell and L. Huff, Mersenne Primes in Imaginary Quadratic Number Fields, (2002).
Eric Weissteins's World of Mathematics, Eisenstein Integer

The actual norms are in A066413.

Cf. A000043, A010527, A057429, A125738, A125739.

maxPi = 3000; primeNormQ[p_] := PrimeQ[1 + 3^p - 2*3^(p/2)*Cos[(p*Pi)/6]]; A066408 = {}; Do[ If[primeNormQ[p = Prime[k]], Print[p]; AppendTo[A066408, p]], {k, 1, maxPi}]; A066408 (* Jean-François Alcover, Oct 21 2011 *)

print1("2, "); /*the only even member; it is special*/ forprime(n=3,2029,if(ispseudoprime(3^n-kronecker(3,n)*3^((n+1)/2)+1),print1(n, ", "))) \\ Serge Batalov, Mar 29 2014

a(26) from Serge Batalov, Mar 29 2014
a(27) from Ryan Propper and Serge Batalov, Jun 18 2023
a(28) from Ryan Propper and Serge Batalov, Jun 20 2023
Corrected link to NMBRTHRY posting. - Serge Batalov, Apr 01 2014

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

Original entry on oeis.org

2, 2, 3, 4, 7, 2, 0, 0, 2, 6, 5, 3, 9, 4, 1, 2, 8, 2, 7, 6, 7, 9, 8, 4, 1, 4, 1, 5, 8, 1, 8, 8, 6, 1, 3, 0, 7, 3, 8, 1, 8, 0, 1, 3, 5, 1, 3, 4, 3, 1, 6, 2, 2, 6, 1, 2, 9, 7, 9, 9, 7, 6, 3, 1, 6, 7, 1, 0, 2, 0, 4, 7, 1, 6, 7, 6, 3, 5, 2, 4, 7, 7, 6, 8, 3, 3, 9, 9, 7, 2, 1, 9, 3, 8, 6, 4, 1, 1, 4, 7, 0, 3, 3, 2, 0

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			22.3472002653941282767984141581886130738180135134316226129799763167102...

Wolfram Alpha, Johnson solid 6

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593.

RealDigits[N[Sqrt[5*(145+58*Sqrt[5]+2*Sqrt[30*(65+29*Sqrt[5])])]/2,200]]

Digits of sqrt(5*(145+58*sqrt(5)+2*sqrt(30*(65+29*sqrt(5)))))/2.

Offset corrected by R. J. Mathar, Aug 15 2010

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Original entry on oeis.org

6, 8, 8, 1, 9, 0, 9, 6, 0, 2, 3, 5, 5, 8, 6, 7, 6, 9, 1, 0, 3, 6, 0, 4, 7, 9, 0, 9, 5, 5, 4, 4, 3, 8, 3, 9, 7, 6, 2, 9, 4, 9, 6, 6, 8, 0, 0, 4, 0, 7, 9, 3, 3, 1, 6, 8, 2, 8, 3, 7, 8, 8, 2, 8, 0, 9, 5, 4, 7, 5, 9, 6, 8, 8, 3, 5, 8, 6, 4, 9, 2, 5, 3, 2, 9, 7, 6, 4, 9, 6

Offset: 0

Paolo Xausa, Jul 29 2024

			0.688190960235586769103604790955443839762949668...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Regular Polygon.
Wikipedia, Apothem.
Index entries for algebraic numbers, degree 4.

Cf. A300074 (circumradius), A375068 (sagitta), A102771 (area).
Cf. apothem of other polygons with unit side length: A020769 (triangle), A020761 (square), A010527 (hexagon), A374971 (heptagon), A174968 (octagon), A375152 (9-gon), A179452 (10-gon), A375191 (11-gon), A375193 (12-gon).
Cf. A019934, A019952, A019863, A131595.

Mathematica
```
First[RealDigits[Cot[Pi/5]/2, 10, 100]]
```

 5/tan(Pi/5) \\ Charles R Greathouse IV, Feb 05 2025

Equals cot(Pi/5)/2 = A019952/2.
Equals 1/(2*tan(Pi/5)) = 1/(2*A019934).
Equals sqrt(1/4 + 1/(2*sqrt(5))).
Equals (1/2)*csc(Pi/5)*cos(Pi/5) = A300074*A019863.
Equals A300074 - A375068.
Equals A131595/30. - Hugo Pfoertner, Jul 30 2024

A088543 Decimal expansion of sqrt(15)/2.

Original entry on oeis.org

1, 9, 3, 6, 4, 9, 1, 6, 7, 3, 1, 0, 3, 7, 0, 8, 4, 4, 2, 5, 8, 9, 6, 3, 2, 6, 9, 9, 8, 9, 1, 1, 9, 9, 8, 0, 5, 4, 1, 6, 4, 6, 0, 8, 5, 2, 6, 4, 5, 7, 9, 5, 4, 1, 3, 2, 9, 3, 7, 8, 6, 8, 8, 3, 0, 5, 6, 7, 4, 1, 5, 4, 5, 9, 6, 8, 4, 8, 9, 5, 1, 6, 7, 5, 9, 6, 4, 3, 6, 8, 8, 4, 2, 9, 3, 3, 6, 7, 5, 8, 9, 5, 8, 1, 5

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Nov 16 2003

This is the ratio of the base of an isosceles triangle to its height when the other two sides are each equal to twice the base.

Exradius opposite the side of length 3 of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. See formulas in the Weisstein link. - Rick L. Shepherd, May 14 2008

			1.93649167310370844258963269989119980541646085264579541329378688305674154596...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Exradius.
Index entries for algebraic numbers, degree 2

Cf. A010527.

Cf. A140239, A140246, A140247, A140248, A140249, A010472.

First[RealDigits[Sqrt[15]/2, 10, 100]] (* Paolo Xausa, Jun 18 2024 *)

sqrt(15)/2 = A010472/2 = A140249/3. - Rick L. Shepherd, May 14 2008

More terms from Rick L. Shepherd, May 14 2008

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

Original entry on oeis.org

3, 6, 7, 5, 5, 2, 5, 9, 6, 9, 4, 7, 8, 6, 1, 3, 6, 6, 3, 4, 0, 8, 8, 4, 3, 3, 2, 2, 0, 8, 6, 4, 6, 2, 9, 4, 2, 6, 4, 9, 2, 4, 3, 2, 0, 2, 4, 4, 4, 2, 7, 1, 0, 1, 8, 6, 6, 2, 4, 4, 0, 1, 3, 5, 2, 7, 3, 5, 3, 5, 3, 5, 6, 4, 6, 1, 7, 9, 8, 6, 3, 2, 2, 6, 9, 2, 0, 0, 1, 9, 2, 1, 5, 4, 4, 7, 2, 5, 9, 4, 7, 1, 7, 9, 8

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

A179589 Decimal expansion of the circumradius of square cupola with edge length 1.

Original entry on oeis.org

1, 3, 9, 8, 9, 6, 6, 3, 2, 5, 9, 6, 5, 9, 0, 6, 7, 0, 2, 0, 3, 1, 5, 4, 0, 5, 3, 9, 4, 3, 1, 9, 9, 8, 7, 6, 4, 6, 7, 3, 5, 2, 2, 5, 6, 3, 8, 6, 6, 2, 3, 8, 8, 7, 9, 9, 3, 0, 9, 3, 6, 3, 2, 3, 1, 5, 0, 3, 7, 3, 5, 9, 2, 0, 3, 7, 9, 8, 0, 2, 9, 9, 1, 1, 4, 8, 2, 8, 3, 0, 0, 5, 0, 1, 4, 4, 6, 8, 0, 3, 0, 4, 2, 9, 4

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			1.398966325965906702031540539431998764673522563866238879930...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588.

Mathematica
```
RealDigits[N[Sqrt[5+2*Sqrt[2]]/2,200]]
```

Digits of sqrt(5+2*sqrt(2))/2.

A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

Original entry on oeis.org

1, 1, 9, 2, 7, 0, 2, 2, 4, 2, 2, 3, 2, 2, 3, 2, 5, 5, 9, 0, 6, 0, 1, 9, 8, 4, 2, 8, 3, 7, 7, 2, 5, 1, 5, 8, 1, 5, 5, 2, 6, 2, 5, 5, 1, 8, 2, 8, 8, 6, 2, 0, 1, 5, 7, 0, 7, 7, 9, 3, 1, 4, 2, 1, 8, 8, 8, 2, 2, 7, 4, 7, 2, 4, 5, 5, 2, 5, 8, 3, 8, 6, 3, 0, 8, 2, 0, 7, 7, 0, 6, 7, 0, 0, 1, 8, 1, 1, 7, 7, 4, 7, 6, 3, 8

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated square pyramid: 9 vertices, 20 edges, and 13 faces.

			1.19270224223223255906019842837725158155262551828862015707793142188822...

Wolfram Alpha, Johnson solid 10

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637.

RealDigits[N[(Sqrt[2]+2*Sqrt[4+3*Sqrt[2]])/6,200]]

Digits of (sqrt(2)+2 sqrt(4+3 sqrt(2)))/6.

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

9, 7, 4, 9, 2, 7, 9, 1, 2, 1, 8, 1, 8, 2, 3, 6, 0, 7, 0, 1, 8, 1, 3, 1, 6, 8, 2, 9, 9, 3, 9, 3, 1, 2, 1, 7, 2, 3, 2, 7, 8, 5, 8, 0, 0, 6, 1, 9, 9, 9, 7, 4, 3, 7, 6, 4, 8, 0, 7, 9, 5, 7, 5, 0, 8, 7, 6, 4, 5, 9, 3, 1, 6, 3, 4, 4, 0, 3, 7, 9, 3, 7, 0, 0, 1, 1, 2, 4, 5, 8, 1, 2, 0, 7, 3, 6, 9, 2, 5, 1, 6, 4, 0, 1, 4

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A375069 Decimal expansion of the sagitta of a regular hexagon with unit side length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A003630 Inert rational primes in Q[sqrt(3)].

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A066408 Numbers n such that the Eisenstein integer (1 - ω)^n - 1 has prime norm, where ω = -1/2 + sqrt(-3)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A375067 Decimal expansion of the apothem (inradius) of a regular pentagon with unit side length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A088543 Decimal expansion of sqrt(15)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A165952 Decimal expansion of 2*sqrt(3)/(3*Pi).

Views

Author

Keywords

Comments

Examples

Links

A165952 Decimal expansion of 2sqrt(3)/(3Pi).