A005471 -id:A005471 - cp's OEIS frontend

A175281 Numbers k such that 6k+1 is a term in A005471.

1, 2, 3, 6, 13, 16, 23, 27, 52, 58, 101, 118, 146, 156, 177, 188, 248, 261, 331, 426, 443, 552, 591, 716, 853, 926, 1248, 1336, 1427, 1521, 1553, 1651, 1752, 1963, 2148, 2502, 2543, 2753, 2883, 3016, 3061, 3152, 3338, 3433, 3481, 4083, 4241, 4511, 4566, 4846

Offset: 1

Zak Seidov, Mar 20 2010

nonn

			k=1: 6k+1 = 7 = A005471(1).
k=2: 6k+1 = 13 = A005471(2).
k=3: 6k+1 = 19 = A005471(3).
k=6: 6k+1 = 37 = A005471(4).

Cf. A005471 (primes of form k^2 + 3*k + 9).

A175282 Positive numbers n with property that n^2+3n+9 is prime (A005471).

Original entry on oeis.org

1, 2, 4, 7, 8, 10, 11, 16, 17, 23, 25, 28, 29, 31, 32, 37, 38, 43, 49, 50, 56, 58, 64, 70, 73, 85, 88, 91, 94, 95, 98, 101, 107, 112, 121, 122, 127, 130, 133, 134, 136, 140, 142, 143, 155, 158, 163, 164, 169, 172, 175, 176, 179, 182, 197, 200, 205, 206, 212, 214, 218

Offset: 1

Zak Seidov, Mar 21 2010

nonn

Notice that at n=-1, n^2+3n+9=7 is also (positive) prime.

Erik Dofs, Solutions of x^3+y^3+z^3=n*x*y*z, Acta Arithm. 73.3 (1995) 201

Cf. A005471, A175281.

A175282 := proc(n)
    option remember;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(a^2+3*a+9) then
                return a;
            end if;
        end do;
    end if;
end proc: # R. J. Mathar, Jun 06 2019

Select[Range[1,400],PrimeQ[ #^2+3*#+9]&]

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

Original entry on oeis.org

10181, 8527, 6967, 5501, 4129, 2851, 1667, 577, -419, -1321, -2129, -2843, -3463, -3989, -4421, -4759, -5003, -5153, -5209, -5171, -5039, -4813, -4493, -4079, -3571, -2969, -2273, -1483, -599, 379, 1451, 2617, 3877, 5231, 6679, 8221, 9857, 11587, 13411, 15329, 17341, 19447, 21647, 31387

Offset: 1

Eric W. Weisstein

Terms are listed in the order of their appearance in sequence b.

This is a transformed version of the polynomial P(x) = 47*x^2 + 9*x - 5209 whose absolute value gives 43 distinct primes for -24 <= x <= 18, found by G. W. Fung in 1988. - Hugo Pfoertner, Dec 13 2019

R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004 (ISBN 0-387-20860-7); see Section A17, p. 59.
Paulo Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag New York, 2004. See p. 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
Carlos Rivera, Problem 12: Prime producing polynomials, The Prime Puzzles & Problems Connection.
Jitender Singh, Prime numbers and factorization of polynomials, arXiv:2411.18366 [math.NT], 2024.
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.

Cf. A002383, A005471, A005846, A007635, A022464, A027753, A027755, A027758, A048059, A050267, A050268, A116206, A117081, A267252.

lst={};Do[p=47*n^2-1701*n+10181;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 29 2009 *)
Select[Table[47n^2-1701n+10181,{n,0,50}],PrimeQ] (* Harvey P. Dale, Oct 03 2011 *)

[n | n <- apply(m->47*m^2-1701*m+10181, [0..100]), isprime(abs(n))] \\ Charles R Greathouse IV, Jun 18 2017

Edited by N. J. A. Sloane, May 10 2007
Further edited by Klaus Brockhaus, Mar 20 2010
More terms (to distinguish from quadratic) from Charles R Greathouse IV, Jun 18 2017

A348720 Decimal expansion of 4cos(2Pi/13)cos(3Pi/13).

Original entry on oeis.org

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

Offset: 1

Peter Bala, Oct 31 2021

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 cubic polynomial has discriminant equal to p^2 and has three real roots, one positive and two negative. Here we consider the positive root in the case a = 1 corresponding to the prime p = 13. See A348721 and A348722 for the negative roots.
Let r_0 = 4*cos(2*Pi/13)*cos(3*Pi/13): r_0 is the positive root of the cubic equation x^3 - x^2 - 4*x - 1 = 0. The negative roots are r_1 = - 4*cos(4*Pi/13)*cos(6*Pi/13) = - 0.2738905549... and r_2 = - 4*cos(8*Pi/13)*cos(12*Pi/13) = - 1.3772028539....
The roots of the cubic are permuted by the linear fractional transformation x -> - 1/(1 + x) of order 3:
r_1 = - 1/(1 + r_0); r_2 = - 1/(1 + r_1); r_0 = - 1/(1 + r_2).
The quadratic mapping z -> z^2 - 2*z - 2 also cyclically permutes the roots. The mapping z -> - z^2 + z + 3 gives the inverse cyclic permutation of the roots.
The algebraic number field Q(r_0) is a totally real cubic field with class number 1 and discriminant equal to 13^2. The Galois group of Q(r_0) over Q is a cyclic group of order 3. See Shanks, Table 1, entry corresponding to a = 1.
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 4 in the table).

			2.651093408937175306253240337787615403132441075705596684018767...

T. W. Cusick and Lowell Schoenfeld, A table of fundamental pairs of units in totally real cubic fields, Math. Comp. 48 (1987), 147-158
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
Index entries for algebraic numbers, degree 3

Cf. A160389, A255240, A255241, A255249, A348721-A348729

evalf(4*cos(2*Pi/13)*cos(3*Pi/13), 100);

RealDigits[4*Cos[2*Pi/13]*Cos[3*Pi/13], 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

polrootsreal(x^3 - x^2 - 4*x - 1)[3] \\ Charles R Greathouse IV, Oct 30 2023

r_0 = 2*(cos(Pi/13) + cos(5*Pi/13)).
r_0 = sin(4*Pi/13)*sin(6*Pi/13) / (sin(2*Pi/13)*sin(3*Pi/13)).
r_0 = Product_{n >= 0} (13*n+4)*(13*n+6)*(13*n+7)*(13*n+9)/( (13*n+2)*(13*n+3)*(13*n+10)*(13*n+11) ).
r_1 = 2*(cos(3*Pi/13) - cos(2*Pi/13)).
r_1 = - sin(Pi/13)*sin(5*Pi/13)/(sin(4*Pi/13)*sin(6*Pi/13)).
r_1 = - Product_{n >= 0} (13*n+1)*(13*n+5)*(13*n+8)*(13*n+12)/( (13*n+4)*(13*n+6)*(13*n+7)*(13*n+9) ).
r_2 = 2*(cos(7*Pi/13) - cos(4*Pi/13)).
r_2 = - sin(2*Pi/13)*sin(3*Pi/13)/(sin(Pi/13)*sin(5*Pi/13)).
r_2 = - Product_{n >= 0} (13*n+2)*(13*n+3)*(13*n+10)*(13*n+11)/( (13*n+1)*(13*n+5)*(13*n+8)*(13*n+12) ).
Equivalently, let z = exp(2*Pi*i/13). Then
r_0 = abs( (1 - z^4)*(1 - z^6)/((1 - z^2)*(1 - z^3)) );
r_1 = - abs( (1 - z)*(1 - z^5)/((1 - z^4)*(1 - z^6)) );
r_2 = - abs( (1 - z^2)*(1 - z^3)/((1 - z)*(1 - z^5)) ).
Note: C = {1, 5, 8, 12} is the subgroup of nonzero cubic residues in the finite field Z_13 with cosets 2*C = {2, 3, 10, 11} and 4*C = {4, 6, 7, 9}.
Equals (-1)^(1/13) + (-1)^(5/13) - (-1)^(8/13) - (-1)^(12/13). - Peter Luschny, Nov 08 2021

A348729 Decimal expansion of the positive root of Shanks's simplest cubic associated with the prime p = 163.

Original entry on oeis.org

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

Offset: 2

Peter Bala, Nov 06 2021

Let a be a natural number 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 = 11, corresponding to the prime p = 163, the three real roots of Shanks' cubic x^3 - 11*x^2 - 14*x - 1 in descending order are r_0 = 12.1582466687..., r_1 = - -0.0759979672... and r_2 = -1.0822487014.... Here we consider the positive root r_1.
The linear fractional transformation z -> - 1/(1 + z) cyclically permutes the three roots r_0, r_1 and r_2: the quadratic mapping z -> z^2 - 12*z - 2 also cyclically permutes the roots.
The algebraic number field Q(r_0) is a totally real cubic field with class number 4 and discriminant equal to 163^2. 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.

			12.15824666871213538260037124700042984524658480470748 ...

D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
Index entries for algebraic numbers, degree 3.

Cf. A005471, A160389, A255240, A255241, A255249, A348720 - A348728.

R := convert([seq(mod(n^3, 163), n = 1..162)], set):
P := k -> sqrt( mul(sin((1/163)*k*r*Pi), r in R) ):
evalf(sqrt(P(3)/P(1)), 105);

rs = Union@Mod[Range[1, 162]^3, 163]; f[k_] := Sqrt[Product[Sin[k*r*Pi/163], {r, rs}]]; RealDigits[Sqrt[f[3]/f[1]], 10, 100][[1]] (* Amiram Eldar, Nov 08 2021 *)

polrootsreal(x^3 - 11*x^2 - 14*x - 1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Let R = {1, 5, 6, 8, ..., 155, 157, 158, 162} denote the multiplicative subgroup of nonzero cubic residues in the finite field Z_163, with cosets 2*R = {2, 7, 9, 10, ..., 153, 154, 156, 161} and 3*R = {3, 4, 11, 14, ..., 149, 152, 159, 160}.
Define P(k) = Product_{r in R, r <= (163-1)/2} sin(k*r*Pi/163). The three roots of the cubic x^3 - 11*x^2 - 14*x - 1 are
r_0 =  sqrt(P(3)/P(1)) = 12.1582466687....
r_1 = -sqrt(P(1)/P(2)) = -0.0759979672....
r_2 = -sqrt(P(2)/P(3)) = -1.0822487014....

A027692 a(n) = n^2 + n + 7.

Original entry on oeis.org

7, 9, 13, 19, 27, 37, 49, 63, 79, 97, 117, 139, 163, 189, 217, 247, 279, 313, 349, 387, 427, 469, 513, 559, 607, 657, 709, 763, 819, 877, 937, 999, 1063, 1129, 1197, 1267, 1339, 1413, 1489, 1567, 1647, 1729, 1813, 1899, 1987, 2077, 2169, 2263, 2359, 2457, 2557

Offset: 0

Patrick De Geest

Integers k for which the discriminant of x^3 - k*x - k is a square. - Jacob A. Siehler, Mar 14 2009
Integers k for which the Galois group of the polynomial x^3 - k*x - k over Q is a cyclic group of order 3. See Conrad, Corollary 2.5. - Peter Bala, Oct 17 2021
From Peter Bala, Nov 18 2021: (Start)
Integers k such that 4*k - 27 is a square.
Integers k for which the Galois group of the polynomial x^3 + k*(x + 1)^2 over Q is the cyclic group C_3 (apply Conrad, Corollary 2.5 and Uchida, Lemma 1).
For the primes in this list see A005471. (End)

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2).
Patrick De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X).
Koji Uchida, Class numbers of cubic cyclic fields, Journal of the Mathematical Society of Japan, Vol. 26, No. 3, 1974, pp. 447-453.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A002522, A005471 (subset of primes), A109007, A176271.

List([0..50],n->n^2+n+7); # Muniru A Asiru, Jul 15 2018

A027692 := proc(n)
    n*(n+1)+7 ;
end proc: # R. J. Mathar, Jun 06 2019

f[n_]:=n^2+n+7;f[Range[0,60]] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2011 *)

a(n)=n^2+n+7 \\ Charles R Greathouse IV, Jun 11 2015

For n > 2: a(n) = A176271(n+1,4). - Reinhard Zumkeller, Apr 13 2010
a(n) = 2*n + a(n-1) (with a(0)=7). - Vincenzo Librandi, Aug 05 2010
G.f.: (-7 + 12*x - 7*x^2)/(x-1)^3. - R. J. Mathar, Feb 06 2011
a(n+1) = n^2 + 3*n + 9, see A005471. - R. J. Mathar, Jun 06 2019
a(n) mod 6 = A109007(n+2). - R. J. Mathar, Jun 06 2019
Sum_{n>=0} 1/a(n) = Pi*tanh(Pi*3*sqrt(3)/2)/(3*sqrt(3)). - Amiram Eldar, Jan 17 2021
From Elmo R. Oliveira, Oct 30 2024: (Start)
E.g.f.: exp(x)*(7 + 2*x + x^2).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A005472 Class numbers of Shanks' simplest cubic fields.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 4, 7, 4, 4, 4, 7, 4, 13, 7, 19, 7, 7, 7, 19, 19, 19, 16, 31, 19, 28, 19, 49, 31, 28, 31, 64, 43, 37, 127, 61, 52, 52, 52, 49, 100, 37, 112, 64, 67, 61, 76, 61, 76, 61, 61, 112, 76, 73, 67, 133, 91, 223, 169, 73, 112, 100, 169, 91, 121, 175

Offset: 1

N. J. A. Sloane

nonn

Class numbers of cubic fields with discriminants p^2, where p runs through the primes in A005471.

All terms are of the form x^2 + 3*y^2 (A003136). - Colin Barker, Nov 30 2014

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..100 from R. J. Mathar).
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152 (see Table 1 page 1140).

Cf. A003136, A005471, A005474.

A175282(n)={
    local(a);
    if(n==1,
        return(1),
        a=A175282(n-1)+1;
        while(1,
            if( isprime(a^2+3*a+9),
                return(a),
                a++
            );
        )
    )
};
A005472(n)={
    local(a,bnf,L,H);
    if(n==1, return(1));
    a=A175282(n);
    bnf=bnfinit(x^3-a*x^2-(a+3)*x-1);
    L=ideallist(bnf,1,2);
    H=bnrclassnolist(bnf,L);
    return(H[1][1]);
};
for(n=1,80, print1(A005472(n)," ") ); /* R. J. Mathar, Jun 06 2019 */

Name edited by Robin Visser, Dec 06 2024

A027722 Numbers k such that k^2+k+7 is a palindrome.

Original entry on oeis.org

0, 1, 17, 31, 177, 274, 280, 301, 313, 1777, 2764, 3001, 27259, 30001, 177237, 300001, 312208, 1762122, 3000001, 27515125, 30000001, 30122098, 300000001, 303758458, 2673533185, 2817818390, 3000000001, 3121001208, 26928832879, 28255878334, 30000000001

Offset: 1

Patrick De Geest

Giovanni Resta, Table of n, a(n) for n = 1..45
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027723, A027692, A027756, A005471, A027729, A027724.

palQ[n_] := Block[{d = IntegerDigits[n]}, d == Reverse[d]]; f[n_] := n^2 + n + 7; Select[Range[0, 10^5], palQ@ f@ # &] (* Giovanni Resta, Aug 29 2018 *)

More terms from Giovanni Resta, Aug 28 2018

A027723 Palindromes of form k^2 + k + 7.

Original entry on oeis.org

7, 9, 313, 999, 31513, 75357, 78687, 90909, 98289, 3159513, 7642467, 9009009, 743080347, 900090009, 31413131413, 90000900009, 97474147479, 3105075705013, 9000009000009, 757082131280757, 900000090000009, 907340818043709, 90000000900000009, 92269201110296229

Offset: 1

Patrick De Geest

From Robert Israel, May 16 2018: (Start)
Palindromes m such that 4*m - 27 is a square.
Each term has an odd number of digits and ends in 3, 7 or 9.
Contains 9*(1+10^k+10^(2*k)) for each k>=1. (End)

Giovanni Resta, Table of n, a(n) for n = 1..45
P. De Geest, Palindromic Quasi_Over_Squares of the form n^2+(n+X)

Cf. A027722, A027692, A027756, A005471, A027721, A027725.

R[1]:= [1,3,5,7,9]: X[1]:= R[1]:
for k from 2 to 6 do
  R[k]:= map(t -> seq(10^(k-1)*j+t,j=0..9),R[k-1]);
X[k]:= map(t -> seq(j+10*t,j=0..9),X[k-1])
od:
Res:= 7,9:
for k from 1 to 6 do
  for j from 1 to 5*10^(k-1) do
      r:= 10^(k+1)*X[k][j]+R[k][j];
      for y from 0 to 9 do
        if issqr(4*(r+10^k*y)-27) then
          x:= r+10^k*y;
          Res:= Res,x;
        fi
od od od:
Res; # Robert Israel, May 16 2018

More terms from Giovanni Resta, Aug 28 2018

A027756 Numbers k such that k^2 + k + 7 is prime.

Original entry on oeis.org

0, 2, 3, 5, 8, 9, 11, 12, 17, 18, 24, 26, 29, 30, 32, 33, 38, 39, 44, 50, 51, 57, 59, 65, 71, 74, 86, 89, 92, 95, 96, 99, 102, 108, 113, 122, 123, 128, 131, 134, 135, 137, 141, 143, 144, 156, 159, 164, 165, 170, 173, 176, 177, 180, 183, 198, 201, 206

Offset: 1

Patrick De Geest

Harvey P. Dale, Table of n, a(n) for n = 1..1000
P. De Geest, World!Of Numbers

Cf. A005471 (associated primes).

[n: n in [0..250] | IsPrime(n^2+n+7)]; // Vincenzo Librandi, Nov 20 2010

Select[Range[0,250],PrimeQ[#^2+#+7]&] (* Harvey P. Dale, Oct 10 2011 *)

is(n)=isprime(n^2+n+7) \\ Charles R Greathouse IV, Feb 20 2017

cp's OEIS Frontend

A175281 Numbers k such that 6k+1 is a term in A005471.

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A175282 Positive numbers n with property that n^2+3n+9 is prime (A005471).

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A050267 Primes or negative values of primes in the sequence b(n) = 47*n^2 - 1701*n + 10181, n >= 0.

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A348720 Decimal expansion of 4*cos(2*Pi/13)*cos(3*Pi/13).

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A348729 Decimal expansion of the positive root of Shanks's simplest cubic associated with the prime p = 163.

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A027692 a(n) = n^2 + n + 7.

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A005472 Class numbers of Shanks' simplest cubic fields.

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A027722 Numbers k such that k^2+k+7 is a palindrome.

A050267 Primes or negative values of primes in the sequence b(n) = 47n^2 - 1701n + 10181, n >= 0.

A348720 Decimal expansion of 4cos(2Pi/13)cos(3Pi/13).