A139851 -id:A139851 - cp's OEIS frontend

A139827 Primes of the form 2x^2 + 2xy + 17y^2.

2, 17, 29, 41, 101, 149, 173, 197, 233, 281, 293, 461, 557, 569, 593, 677, 701, 761, 809, 821, 857, 941, 953, 1097, 1217, 1229, 1289, 1361, 1481, 1493, 1553, 1601, 1613, 1733, 1877, 1889, 1913, 1949, 1997, 2081, 2129, 2141, 2153, 2213, 2273, 2309, 2393, 2417

Offset: 1

T. D. Noe, May 02 2008, May 07 2008

Discriminant = -132.
Consider the quadratic form f(x,y) = ax^2 + bxy + cy^2. When the discriminant d=b^2-4ac is -4 times an idoneal number (A000926), there is exactly one class for each genus. As a result, the primes generated by f(x,y) are the same as the primes congruent to S (mod -d), where S is a set of numbers less than -d. The table on page 60 of Cox shows that there are exactly 331 quadratic forms having this property. The 217 sequences starting with this one complete the collection in the OEIS.
When a=1 and b=0, f(x,y) is a quadratic form whose congruences are discussed in A139642. Let N be an idoneal number. Then there are 2^r reduced quadratic forms whose discriminant is -4N, where r=1,2,3, or 4. By collecting the residuals p (mod 4N) for primes p generated by the i-th reduced quadratic form, we can empirically find a set Si. To show that the 2^r sets Si are complete, we only need to show that the union of the Si is equal to the set of numbers k such that the Jacobi symbol (-k/4N)=1.

David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A139643, A139841-A139843 (d=-408), A139644, A139844-A139850 (d=-420), A139645, A139851-A139853 (d=-448), A139502, A139854-A139860 (d=-480), A139646, A139861-A139863 (d=-520), A139647, A139864-A139866 (d=-532), A139648, A139867-A139873 (d=-660), A139506, A139874-A139880 (d=-672), A139649, A139881-A139883 (d=-708), A139650, A139884-A139886 (d=-760), A139651, A139887-A139893 (d=-840), A139652, A139894-A139896 (d=-928), A139502, A139855, A139857, A139858, A139897-A139899, A139902 (d=-960).
Cf. also  A139653, A139904-A139906 (d=-1012), A139654, A139907-A139913 (d=-1092), A139655, A139914-A139920 (d=-1120), A139656, A139921-A139927 (d=-1248), A139657, A139928-A139934 (d=-1320), A139658, A139935-A139941 (d=-1380), A139659, A139942-A139948 (d=-1428), A139660, A139949-A139955 (d=-1540), A139661, A139956-A139962 (d=-1632), A139662, A139963-A139969 (d=-1848), A139663, A139970-A139976 (d=-2080), A139664, A139977-A139983 (d=-3040), A139665, A139984-A139998 (d=-3360), A139666, A139999-A140013 (d=-5280), A139667, A140014-A140028 (d=-5460), A139668, A140029-A140043 (d=-7392).
For a more complete list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

[ p: p in PrimesUpTo(2500) | p mod 132 in {2, 17, 29, 41, 65, 101}]; // Vincenzo Librandi, Jul 29 2012

QuadPrimes2[2, -2, 17, 2500] (* see A106856 *)
t = Table[{2, 17, 29, 41, 65, 101} + 132*n, {n, 0, 50}]; Select[Flatten[t], PrimeQ] (* T. D. Noe, Jun 21 2012 *)

v=[2, 17, 29, 41, 65, 101]; select(p->setsearch(v,p%132),primes(100)) \\ Charles R Greathouse IV, Jan 08 2013

The primes are congruent to {2, 17, 29, 41, 65, 101} (mod 132).

A261521 a(n) = n^2 + 2*n + 29.

Original entry on oeis.org

29, 32, 37, 44, 53, 64, 77, 92, 109, 128, 149, 172, 197, 224, 253, 284, 317, 352, 389, 428, 469, 512, 557, 604, 653, 704, 757, 812, 869, 928, 989, 1052, 1117, 1184, 1253, 1324, 1397, 1472, 1549, 1628, 1709, 1792, 1877, 1964, 2053, 2144, 2237, 2332, 2429, 2528

Offset: 0

Jake Saville, Oct 02 2015

A139851, which lists primes of the form 4x^2 + 4xy + 29y^2, contains all prime values of a(n). - Altug Alkan, Oct 02 2015

			For n = 3, a(3) = 3^2 + 2*3 + 29 = 44.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005408, A005563, A139851.

[n^2+2*n+29: n in [0..50]]; // Vincenzo Librandi, Oct 03 2015

Table[n^2 + 2 n + 29, {n, 0, 50}] (* Bruno Berselli, Oct 25 2015 *)
LinearRecurrence[{3,-3,1},{29,32,37},50] (* Harvey P. Dale, Oct 14 2023 *)

vector(50, n, n--; n^2+2*n+29) \\ Altug Alkan, Oct 02 2015

Vec((29 - 55*x + 28*x^2)/(1-x)^3 + O(x^100)) \\ Altug Alkan, Oct 17 2015

Python
```
def a(x):return x*x+2*x+27
```

a(n) = a(n-1) + A005408(n), a(0) = 29, for n > 0. - Altug Alkan, Oct 02 2015
From Vincenzo Librandi, Oct 03 2015: (Start)
G.f.: (29 - 55*x + 28*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
a(n) = A005563(n) + 29. - Omar E. Pol, Oct 17 2015
E.g.f.: (29 + 3*x + x^2)*exp(x). - Elmo R. Oliveira, Oct 19 2024

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A139827 Primes of the form 2x^2 + 2xy + 17y^2.

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A261521 a(n) = n^2 + 2*n + 29.

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