A175282 -id:A175282 - cp's OEIS frontend

A005471 Primes of the form m^2 + 3m + 9, where m can be positive or negative.

7, 13, 19, 37, 79, 97, 139, 163, 313, 349, 607, 709, 877, 937, 1063, 1129, 1489, 1567, 1987, 2557, 2659, 3313, 3547, 4297, 5119, 5557, 7489, 8017, 8563, 9127, 9319, 9907, 10513, 11779, 12889, 15013, 15259, 16519, 17299, 18097, 18367, 18913, 20029

Offset: 1

N. J. A. Sloane

Primes of the form m^2 + m + 7, for some m >= 0. - Daniel Forgues, Jan 26 2020
Primes p such that 4*p - 27 is a square. Also, primes p such that the Galois group of the polynomial X^3 - p*X + p over Q is the cyclic group of order 3. See Conrad, Corollary 2.5. - Peter Bala, Oct 17 2021
From Peter Bala, Nov 18 2021: (Start)
Primes p such that the Galois group of the cubic X^3 + p*(X + 1)^2 over Q is the cyclic group C_3.
If p = m^2 + 3*m + 9 is prime then the Galois group of the cubic X^3 - m*X^2 - (m + 3)*X - 1 over Q is C_3. See Shanks.
The pair of cubics X^3 - m*p*X^2 - 3*(m+1)*p*X - (2*m+3)*p and X^3 - 2*p*X^2 + p*(p - 10)*X + p*(p - 8) also have their Galois groups over Q equal to C_3 (both cubics are irreducible over Q by Eisenstein's criteria). Apply Conrad, Corollary 2.5. (End)

			For m = -11, -10, ..., 22 the primes of the form m^2+3m+9 are 97, 79, 37, 19, 13, 7, 7, 13, 19, 37, 79, 97, 139, 163, 313, 349.

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

T. D. Noe, Table of n, a(n) for n = 1..1000
S. Barbero, U. Cerruti, and N. Murru, Identities Involving Zeros of Ramanujan and Shanks Cubic Polynomials, J. Integer Seq., Vol. 16 (2013), Article 13.8.1.
Keith Conrad, Galois groups of cubics and quartics (not in characteristic 2)
Hyun Kwang Kim and Jung Soo Kim, Evaluation of zeta function of the simplest cubic field at negative odd integers, Math. Comp. 71 (2002), no. 239, 1243-1262.
D. Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.
Koji Uchida, Class numbers of cubic cyclic fields, J. Math. Soc. Japan, Vol. 26, No. 3, 1974, pp. 447 - 453.

Primes in A027692.

Cf. A175282, A227622, A349461.

[a: n in [-1..150] | IsPrime(a) where a is  n^2+3*n+9]; // Vincenzo Librandi, Mar 22 2013

A005471 := proc(n)
    if n = 1 then
        7;
    else
        A175282(n-1)*(3+A175282(n-1))+9 ;
    end if;
end proc: # R. J. Mathar, Jun 06 2019

Select[Table[n^2 + 3*n + 9, {n, -1, 200}], PrimeQ] (* T. D. Noe, Mar 21 2013 *)

a(n) == 1 (mod 6). - Zak Seidov, Mar 20 2010

a(n+1) = A175282(n)^2 + 3*A175282(n) + 9. - R. J. Mathar, Jun 06 2019

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

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

Zak Seidov, Mar 21 2010

nonn

Or, primes in A175282.

Cf. A005471, A175281, A175282.

[ n: n in [0..1250] | IsPrime(n) and IsPrime(n^2+3*n+9)] // Vincenzo Librandi, Jan 30 2011

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

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A005471 Primes of the form m^2 + 3m + 9, where m can be positive or negative.

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A005472 Class numbers of Shanks' simplest cubic fields.

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

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