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A002476 Primes of form 6m + 1.
(Formerly M4344 N1819)
133
7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Equivalently, primes of the form 3m + 1.

Primes p dividing sum(k = 0, p, C(2k, k)) - 3 = A006134(p) - 3 - Benoit Cloitre, Feb 08 2003

Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003

Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014

Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005

Primes p such that p^2 divides Sum_{m = 1 .. 2(p - 1)} Sum_{k = 1 .. m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006

A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007

The set of primes of the form 3n + 1 is a superset of the set of greater of twin primes larger than five (A006512). - Paul Muljadi, Jun 05 2008

Also primes p such that the arithmetic mean of divisors of p^2 is an integer : sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008

Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012

Is this the same sequence as A139492? No, because 13 is in this sequence but not in A139492. Is A139492 a subsequence of A002476? - Jon Perry, Nov 25 2013

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.

David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8

K. G. Reuschle, Tafeln Complexer Primzahlen. K\"{o}nigl. Akademie der Wissenschaften, Berlin, 1875, p. 1.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

C. Banderier, Calcul de (-3/p)

A. Granville and G. Martin, Prime number races

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

FORMULA

Sum_{n >= 1} 1/a(n)^2 = A175644. Sum_{n >= 1} 1/a(n)^3 = A175645. - R. J. Mathar, Apr 03 2011

EXAMPLE

Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3))).

Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.

17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

MAPLE

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

MATHEMATICA

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

PROG

(MAGMA) [n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

(PARI) select(p->p%3==1, primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

(Haskell)

a002476 n = a002476_list !! (n-1)

a002476_list = filter ((== 1) . (`mod` 6)) a000040_list

-- Reinhard Zumkeller, Jan 15 2013

CROSSREFS

Cf. A045331, A242660.

For values of n see A024899. Primes of form 3n - 1 give A003627.

These are the primes arising in A024892, A024899, A034936.

A091178 gives prime index.

Cf. A006512.

Subsequence of A016921.

Cf. A004611 (multiplicative closure).

Sequence in context: A107925 * A123365 A144921 A040079 A038160 A106870

Adjacent sequences:  A002473 A002474 A002475 * A002477 A002478 A002479

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 25 23:29 EDT 2014. Contains 248566 sequences.