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.

Previous Showing 21-28 of 28 results.

A348724 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.

Original entry on oeis.org

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

Views

Author

Peter Bala, Oct 31 2021

Keywords

Comments

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.
In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348723 and A348725.
Here we consider the absolute value of the root r_1. In Cusick and Schoenfeld r_1 is denoted by E_2. See case 9 in the table.

Examples

			0.22187616226319093426668005018505061559919549440775...

Links

Crossrefs

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

Programs

  • Maple
    evalf(sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)), 100);
  • Mathematica
    RealDigits[Sin[2*Pi/19]*Sin[3*Pi/19]*Sin[5*Pi/19]/(Sin[4*Pi/19]*Sin[6*Pi/19]*Sin[9*Pi/19]), 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19)*sin(9*Pi/19)) \\ Michel Marcus, Nov 08 2021

Formula

|r_1| = 2*(cos(3*Pi/19) + cos(5*Pi/19) - cos(2*Pi/19)) - 1.
|r_1| = sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)/(sin(4*Pi/19)*sin(6*Pi/19) *sin(9*Pi/19)) = 1/(8*cos(2*Pi/19)*cos(3*Pi/19)*cos(5*Pi/19)).
|r_1| = Product_{n >= 0} (19*n+2)*(19*n+3)*(19*n+5)*(19*n+14)*(19*n+16)*(19*n+17)/( (19*n+4)*(19*n+6)*(19*n+9)*(19*n+10)*(19*n+13)*(19*n+15) ).
Let z = exp(2*Pi*i/19). Then
|r_1| = abs( (1 - z^2)*(1 - z^3)*(1 - z^5)/((1 - z^4)*(1 - z^6)*(1 - z^9)) ).
Note: C = {1, 7, 8, 11, 12, 18} is the subgroup of nonzero cubic residues in the finite field Z_19 with cosets 2*C = {2, 3, 5, 14, 16, 17} and 4*C = {4, 6, 9, 10, 13, 15}.
Equals -1 - (-1)^(2/19) + (-1)^(3/19) + (-1)^(5/19) - (-1)^(14/19) - (-1)^(16/19) + (-1)^(17/19). - Peter Luschny, Nov 08 2021

A348725 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 19.

Original entry on oeis.org

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

Views

Author

Peter Bala, Oct 31 2021

Keywords

Comments

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1.
In the case a = 2, corresponding to the prime p = 19, Shanks' cyclic cubic is x^3 - 2*x^2 - 5*x - 1 of discriminant 19^2. The polynomial has three real roots, one positive and two negative. Let r_0 = 3.507018644... denote the positive root. The other roots are r_1 = - 1/(1 + r_0) = - 0.2218761622... and r_2 = - 1/(1 + r_1) = - 1.2851424818.... See A348723 (r_0) and A348724 (|r_1|).
Here we consider the absolute value of the root r_2.

Examples

			1.28514248182978536439411987353062741342678092572261 ...

Links

Crossrefs

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

Programs

  • Maple
    evalf(sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)/(sin(2*Pi/19)*sin(3*Pi/19)*sin(5*Pi/19)), 100);
  • Mathematica
    RealDigits[Sin[Pi/19]*Sin[7*Pi/19]*Sin[8*Pi/19]/(Sin[2*Pi/19]*Sin[3*Pi/19]*Sin[5*Pi/19]), 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

Formula

|r_2| = sin(Pi/19)*sin(7*Pi/19)*sin(8*Pi/19)/(sin(2*Pi/19)*sin(3*Pi/19)* sin(5*Pi/19)) = 1/(8*cos(Pi/19)*cos(7*Pi/19)*cos(8*Pi/19)).
|r_2| = Product_{n >= 0} (19*n+1)*(19*n+7)*(19*n+8)*(19*n+11)*(19*n+12)*(19*n+18)/ ( (19*n+2)*(19*n+3)*(19*n+5)*(19*n+14)*(19*n+16)*(19*n+17) ).
|r_2| = 2*(cos(Pi/19) + cos(7*Pi/19) - cos(8*Pi/19)) - 1.
Let z = exp(2*Pi*i/19). Then
|r_2| = abs( (1 - z)*(1 - z^7)*(1 - z^8)/((1 - z^2)*(1 - z^3)*(1 - z^5)) ).
Note: C = {1, 7, 8, 11, 12, 18} is the subgroup of nonzero cubic residues in the finite field Z_19 with cosets 2*C = {2, 3, 5, 14, 16, 17} and 4*C = {4, 6, 9, 10, 13, 15}.
Equals -1 + (-1)^(1/19) + (-1)^(7/19) - (-1)^(8/19) + (-1)^(11/19) - (-1)^(12/19) - (-1)^(18/19). - Peter Luschny, Nov 08 2021

A348726 Decimal expansion of the positive root of Shanks' simplest cubic associated with the prime p = 37.

Original entry on oeis.org

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

Views

Author

Peter Bala, Oct 31 2021

Keywords

Comments

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The polynomial has three real roots, one positive and two negative.
In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.3447123654..., r_1 = - 0.1576115578... and r_2 = - 1.1871008076.... Here we consider the positive root r_0. See A348727 (|r_1|) and A348728 (|r_2|) for the other two roots.
The algebraic number field Q(r_0) is a totally real cubic field with class number 1 and discriminant equal to 37^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. The real numbers r_0 and 1 + r_0 are two independent fundamental units of the field Q(r_0). See Shanks. In Cusick and Schoenfeld, r_0 and r_1 (denoted there by E_1 and E_2) are taken as a fundamental pair of units (see case 37 in the table).
Let R = <1, 6, 8, 10, 11, 14, 23, 26, 27, 29, 31, 36> denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_37, with cosets 2*R = {2, 9, 12, 15, 16, 17, 20, 21, 22, 25, 28, 35} and 3*R = {3, 4, 5, 7, 13, 18, 19, 24, 30, 32, 33, 34}.
Define R(k) = sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*sin(10*k*Pi/37)* sin(11*k*Pi/37)*sin(14*k*Pi/37). Then the three roots of the cubic x^3 - 4*x^2 - 7*x - 1 are
r_0 = - R(2)/R(3) = 5.3447123654..., r_1 = - R(1)/R(2) = - 0.1576115578... and r_2 = R(3)/R(1) = - 1.1871008076....
The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots of the cubic polynomial.
The quadratic mapping z -> z^2 - 5*z - 2 also cyclically permutes the roots of the cubic: the inverse cyclic permutation of the roots is given by z -> - z^2 + 4*z + 6.

Examples

			5.34471236545183496316569141884698646995869587081422 ...

Links

Crossrefs

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

Programs

  • Maple
    R := k -> sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*
sin(10*k*Pi/37)*sin(11*k*Pi/37)*sin(14*k*Pi/37): evalf(-R(2)/R(3), 100);
  • Mathematica
    f[ks_,m_] := Product[Sin[k*Pi/m], {k,ks}]; ks = {1, 6, 8, 10, 11, 14}; RealDigits[f[2*ks,37]/f[3*ks,37], 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

Formula

r_0 = 1 + 2*(cos(3*Pi/37) - cos(4*Pi/37) + cos(5*Pi/37) + cos(7*Pi/37) + cos(13*Pi/37) - cos(18*Pi/37)).
r_0 = |R(2)/R(3)| = Product_{n >= 0} ( Product_{k in the coset 2*R} (37*n+k) )/( Product_{k in the coset 3*R} (37*n + k) );
|r_1| = |R(1)/R(2)| = Product_{n >= 0} ( Product_{k in the group R} (37*n+k) )/( Product_{k in the coset 2*R} (37*n + k) );
|r_2| = |R(3)/R(1)| = Product_{n >= 0} ( Product_{k in the coset 3*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).
R(2)/R(1) + R(2)/R(3) = 1 = R(3)/R(2) - R(3)/R(1) = R(1)/R(2) - R(1)/R(3).

A348727 Decimal expansion of the absolute value of one of the negative roots of Shanks' simplest cubic associated with the prime p = 37.

Original entry on oeis.org

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

Views

Author

Peter Bala, Oct 31 2021

Keywords

Comments

Let a be an integer and let p be a prime of the form a^2 + 3*a + 9 (see A005471). Shanks introduced a family of cyclic cubic fields generated by the roots of the polynomial x^3 - a*x^2 - (a + 3)*x - 1. The polynomial has three real roots, one positive and two negative.
In the case a = 4, corresponding to the prime p = 37, the three real roots of the cubic x^3 - 4*x^2 - 7*x - 1 in descending order are r_0 = 5.344712365..., r_1 = - 0.1576115578... and r_2 = - 1.187100807.... Here we consider the absolute value of the root r_1 (|E_2| in the notation of Cusick and Schoenfeld). See A348726 (r_0) and A348728 (|r_2|) for the other two roots.

Examples

			0.15761155784542576148232132012422537060584871913055 ...

Links

Crossrefs

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348729.

Programs

  • Maple
    R := k -> sin(k*Pi/37)*sin(6*k*Pi/37)*sin(8*k*Pi/37)*sin(10*k*Pi/37)* sin(11*k*Pi/37)*sin(14*k*Pi/37): evalf(R(1)/R(2), 100);
  • Mathematica
    f[ks_,m_] := Product[Sin[k*Pi/m], {k,ks}]; ks = {1, 6, 8, 10, 11, 14}; RealDigits[f[ks,37]/f[2*ks,37], 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

Formula

|r_1| = 1/((2^6)*cos(Pi/37)*cos(6*Pi/37)*cos(8*Pi/37)*cos(10*Pi/37)*cos(11*Pi/37)* cos(14*Pi/37)).
|r_1| = 2*(cos(2*Pi/37) - cos(9*Pi/37) + cos(12*Pi/37) - cos(15*Pi/37) + cos(16*Pi/37) - cos(17*Pi/37)) - 1.
|r_1| = R(1)/R(2), where R(k) = sin(k*Pi/37)*sin(6*k*Pi/37)* sin(8*k*Pi/37)*sin(10*k*Pi/37)*sin(11*k*Pi/37)*sin(14*k*Pi/37).
Let R = <1, 6, 8, 10, 11, 14, 23, 26, 27, 29, 31, 36> denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_37, with cosets 2*R = {2, 9, 12, 15, 16, 17, 20, 21, 22, 25, 28, 35} and 3*R = {3, 4, 5, 7, 13, 18, 19, 24, 30, 32, 33, 34}. Then constant equals
Product_{n >= 0} ( Product_{k in the coset 2*R} (37*n+k) )/( Product_{k in the group R} (37*n + k) ).

A175283 Numbers k with the property that k and k^2 + 3k+9 are primes.

Original entry on oeis.org

2, 7, 11, 17, 23, 29, 31, 37, 43, 73, 101, 107, 127, 163, 179, 197, 239, 277, 281, 317, 331, 359, 367, 421, 457, 463, 487, 541, 569, 613, 617, 619, 661, 709, 739, 773, 787, 809, 823, 877, 941, 947, 953, 991, 1019, 1031, 1033, 1039, 1051, 1087, 1163, 1187
Offset: 1

Views

Author

Zak Seidov, Mar 21 2010

Keywords

Comments

Or, primes in A175282.

Crossrefs

Cf. A005471, A175281, A175282.

Programs

  • Magma
    [ n: n in [0..1250] | IsPrime(n) and IsPrime(n^2+3*n+9)] // Vincenzo Librandi, Jan 30 2011
  • Mathematica
    Select[Prime[Range[400]],PrimeQ[ #^2+3*#+9]&]

A175284 Primes p of the form p=n^2+3n+9 such that q=p^2+3p+9 is also prime.

Original entry on oeis.org

7, 37, 163, 709, 877, 46447, 67867, 81517, 118687, 238639, 292147, 331207, 430999, 497737, 548347, 628063, 1120429, 1412539, 1462897, 1655089, 1680919, 1955809, 2642257, 3205897, 3358063, 3394813, 3781087, 4654813, 4715419, 4745869
Offset: 1

Views

Author

Zak Seidov, Mar 21 2010

Keywords

Comments

Intersection of A175283 and A005471.

Crossrefs

Cf. A005471, A175281-3.

Programs

  • Mathematica
    Reap[Do[If[PrimeQ[p=n^2+3n+9]&&PrimeQ[q=p^2+3p+9],Sow[p]],{n,-1,10^4}]][[2,1]]

A175285 Numbers n with property that 42*n+37 is in A175284.

Original entry on oeis.org

0, 3, 16, 20, 1105, 1615, 1940, 2825, 5681, 6955, 7885, 10261, 11850, 13055, 14953, 26676, 33631, 34830, 39406, 40021, 46566, 62910, 76330, 79953, 80828, 90025, 110828, 112271, 112996, 116340, 126116, 153065, 165126, 175828, 205030, 218953
Offset: 0

Views

Author

Zak Seidov, Mar 21 2010

Keywords

Comments

Except of the first term in A175284, all terms == 37 mod 42.

Crossrefs

Cf. A005471, A175281-4.

A267378 Primes of the form k^4 - k^2 + 7.

Original entry on oeis.org

7, 19, 79, 607, 9907, 20599, 65287, 104659, 129967, 331207, 1047559, 1184839, 1872799, 3746167, 4098607, 6762607, 7308919, 11313139, 20146639, 21376759, 28392919, 43040167, 54693427, 59961799, 84925447, 104050207, 108232819, 131068159, 168883027, 187375039
Offset: 1

Views

Author

Emre APARI, Jan 13 2016

Keywords

Comments

From Robert Israel, Jan 18 2016: (Start)
Subset of A005471.
All terms == 7 or 19 (mod 30). (End)

Examples

			k = 5: 5^4-5^2+7=607 (is prime).

Crossrefs

Cf. A005471.

Programs

  • Magma
    [a: n in [1..150] | IsPrime(a) where a is n^4-n^2+7]; // Vincenzo Librandi, Jan 15 2016
  • Maple
    select(isprime, [seq(x^4-x^2+7,x=1..1000)]); # Robert Israel, Jan 18 2016
  • Mathematica
    Select[Table[k^4 - k^2 + 7, {k, 100}], PrimeQ] (* Michael De Vlieger, Jan 13 2016 *)
  • PARI
    lista(nn) = for(n=1, nn, if(ispseudoprime(p=n^4-n^2+7), print1(p, ", "))); \\ Altug Alkan, Jan 15 2016

Extensions

More terms from Vincenzo Librandi, Jan 15 2016
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