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A002145 Primes of form 4n+3.
(Formerly M2624 N1039)
175
3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Or, odd primes p such that -1 is not a square mod p, i.e., the Legendre symbol (-1/p) = -1. [LeVeque I, p. 66]. - N. J. A. Sloane, Jun 28 2008

Primes which are not the sum of two squares, see the comment in A022544. - Artur Jasinski, Nov 15 2006

Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)

sin(a(n)*Pi/2) = -1 with Pi = 3.1415..., see A070750. - Reinhard Zumkeller, May 04 2002. (Misleading in the sense that the formula characterizes the super-sequence A004767. - R. J. Mathar, Jul 28 2014)

Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1. - Benoit Cloitre, Oct 22 2002

For p and q both belonging to the sequence, exactly one of the congruences x^2 = p (mod q), x^2 = q (mod p) is solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003

Also primes p that divide L((p-1)/2) or L((p+1)/2), where L(n) = A000032(n), the Lucas numbers. Union of A122869 and A122870. - Alexander Adamchuk, Sep 16 2006

Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - Alexander Adamchuk, Nov 30 2006

Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - Alexander Adamchuk, Apr 18 2007

This sequence is a proper subset of the set of the absolute values of negative fundamental discriminants (A003657). - Paul Muljadi, Mar 29 2008

Bernard Frénicle de Bessy discovered that such primes cannot be the hypothenuse of a a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after Paul Curtz, Sep 10 2008

A079261(a(n)) = 1; complement of A145395. - Reinhard Zumkeller, Oct 12 2008

Subsequence of A007970. - Reinhard Zumkeller, Jun 18 2011

A151763(a(n)) = -1.

Primes p such that p XOR 2 = p - 2. Brad Clardy, Oct 25 2011 (Misleading in the sense that this is a formula for the super-sequence A004767. - R. J. Mathar, Jul 28 2014)

It appears that each term of A004767 is the mean of two terms of this subsequence of primes therein; cf A245203. - M. F. Hasler, Jul 13 2014

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.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 252.

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 66.

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 and Moshe Levin, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

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].

D. Alpern, Gaussian primes

A. Granville and G. Martin, Prime number races

H. J. Smith, Gaussian Primes

Eric Weisstein's World of Mathematics, Gaussian Prime

Eric Weisstein's World of Mathematics, "Gaussian Integer".

Wolfram Research, The Gauss Reciprocity Law

Index entries for Gaussian integers and primes

FORMULA

Remove from A000040 terms that are in A002313.

Intersection of A000040 and A004767. - Alonso del Arte, Apr 22 2014

MAPLE

A002145 := proc(n)

    option remember;

    if n = 1 then

        3;

    else

        a := nextprime(procname(n-1)) ;

        while a mod 4 <>  3 do

            a := nextprime(a) ;

        end do;

        return a;

    end if;

end proc:

seq(A002145(n), n=1..20) ; # R. J. Mathar, Dec 08 2011

MATHEMATICA

Select[4Range[150] - 1, PrimeQ] (* Alonso del Arte, Dec 19 2013 *)

Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] == 1 &] (* or *)

Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == -1 &] (* Robert G. Wilson v, May 11 2014 *)

PROG

(PARI) forprime(p=2, 1e3, if(p%4==3, print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

(Haskell)

a002145 n = a002145_list !! (n-1)

a002145_list = filter ((== 3) . (`mod` 4)) a000040_list

-- Reinhard Zumkeller, Sep 23 2011

(MAGMA) [4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // Arkadiusz Wesolowski, Nov 15 2013

(Sage)

A002145_list = lambda n: filter(lambda p: p % 4 == 3, list(primes(1, n)))  # Peter Luschny, Jul 29 2014

CROSSREFS

Cf. A002144, A122869, A122870, A000032, A003657. Apart from initial term, same as A045326.

Cf. A016105.

Cf. A004614 (multiplicative closure).

Sequence in context: A080978 A160216 A181516 * A092109 A117991 A118260

Adjacent sequences:  A002142 A002143 A002144 * A002146 A002147 A002148

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from James A. Sellers, Apr 21 2000

STATUS

approved

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Last modified October 23 03:24 EDT 2014. Contains 248411 sequences.