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A068228 Primes congruent to 1 (mod 12). 98
13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349, 373, 397, 409, 421, 433, 457, 541, 577, 601, 613, 661, 673, 709, 733, 757, 769, 829, 853, 877, 937, 997, 1009, 1021, 1033, 1069, 1093, 1117, 1129, 1153, 1201, 1213, 1237, 1249, 1297 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This has several equivalent definitions (cf. the Tunnell link)

Also primes of the form x^2 + 9y^2 (discriminant 36). - T. D. Noe, May 07 2005

Also primes of the form x^2 + 12y^2 (discriminant 48). Cf. A140633. - T. D. Noe, May 19 2008

Also primes of the form x^2 + 4*x*y + y^2.

Also primes of the form x^2 + 2*x*y - 2*y^2 (cf. A084916).

Also primes of the form x^2 + 6*x*y - 3*y^2.

Also primes of the form 4*x^2 + 8*x*y + y^2.

Also primes of the form u^2 - 3v^2 (use the transformation {u,v} = {x+2y,y}). - Tito Piezas III, Dec 28 2008

Sequence lists generalized cuban primes (A007645) that are the sum of 2 nonzero squares. - Altug Alkan, Nov 25 2015

REFERENCES

Z. I. Borevich and I. R. Shafarevich, Number Theory. Academic Press, NY, 1966.

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

D. B. Zagier, Zetafunktionen und quadratische Koerper, Springer, 1981.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

William C. Jagy and Irving Kaplansky, Positive definite binary quadratic forms that represent the same primes [Cached copy, with permission]

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

J. B. Tunnell, Proofs of Conjectures Concerning Entry A033212, the Sequence of Primes Congruent to 1 or 19 (mod 30)

J. Voight, Quadratic forms that represent almost the same primes, Math. Comp., Vol. 76 (2007), pp. 1589-1617.

MAPLE

select(isprime, [seq(i, i=1..10000, 12)]); # Robert Israel, Nov 27 2015

MATHEMATICA

Select[Prime/@Range[250], Mod[ #, 12]==1&]

Select[Range[13, 10^4, 12], PrimeQ] (* Zak Seidov, Mar 21 2011 *)

PROG

(PARI) for(i=1, 250, if(prime(i)%12==1, print(prime(i))))

(PARI) forstep(p=13, 10^4, 12, isprime(p)&print(p)); \\ Zak Seidov, Mar 21 2011

(MAGMA) [p: p in PrimesUpTo(1400) | p mod 12 in {1}]; // Vincenzo Librandi, Jul 14 2012

For other programs see the "Binary Quadratic Forms and OEIS" link.

CROSSREFS

Cf. A068227, A068229, A040117, A068231, A068232, A068233, A068234, A068235, A139643, A141122, A140633, A264732.

Subsequence of A084916.

Subsequence of A007645.

Also primes in A084916, A020672.

Cf. A141123 (d=12), A068228 (Primes congruent to 1 (mod 12)) A141111, A141112 (d=65).

Cf. A141187 (d=48) A038872 (d=5), A038873 (d=8), A068228, A141123 (d=12), A038883 (d=13), A038889 (d=17), A141111, A141112 (d=65).

For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.

Sequence in context: A140112 A089030 A141122 * A031339 A034938 A139530

Adjacent sequences:  A068225 A068226 A068227 * A068229 A068230 A068231

KEYWORD

easy,nonn

AUTHOR

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

EXTENSIONS

Edited by Dean Hickerson, Feb 27 2002

Entry revised by N. J. A. Sloane, Oct 18 2014 (Edited, merged with A141122, submitted by Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 05 2008).

STATUS

approved

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Last modified February 6 01:00 EST 2016. Contains 268029 sequences.