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A002476
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Primes of form 6m + 1.
(Formerly M4344 N1819)
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114
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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
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OFFSET
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1,1
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COMMENTS
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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 + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that p^2 divides Sum[Sum[(2k)!/(k!)^2, {k, 1, m}], {m, 1, 2(p - 1)}]. - 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) [From Ctibor O. Zizka, Sep 15 2008]
Fermat knew that these numbers can also be expressed as x + 3y^2 and are therefore not prime in Z[sqrt(-3)]. - Alonso del Arte, Dec 07 2012
Is this the same sequence as A139492?
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REFERENCES
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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 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).
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LINKS
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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
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FORMULA
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Sum_{n >= 1} 1/a(n)^2 = A175644. Sum_{n >= 1} 1/a(n)^3 = A175645. - R. J. Mathar, Apr 03 2011
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EXAMPLE
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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.
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MAPLE
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a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];
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MATHEMATICA
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Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)
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PROG
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(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
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CROSSREFS
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Cf. A045331.
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.
Sequence in context: A107925 * A123365 A144921 A040079 A038160 A106870
Adjacent sequences: A002473 A002474 A002475 * A002477 A002478 A002479
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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