

A002145


Primes of form 4n+3.
(Formerly M2624 N1039)


168



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
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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.  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
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((p1)/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 ((p1)!! + 1) or ((p2)!! + 1).  Alexander Adamchuk, Nov 30 2006
Also odd primes p that divide ((p1)!!  1) or ((p2)!!  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
Frenicle discovered these terms from A002144 as missing in A000040(n+1). A002144 and A002145 are companions. See A102261 (2, 6, 6) . He also mentioned primes of the form 4n  1. [From Paul Curtz, Sep 10 2008]
A079261(a(n)) = 1; complement of A145395. [From Reinhard Zumkeller, Oct 12 2008]
Primes p such that arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes, this one and A002144. [From Ctibor O. Zizka, Oct 20 2009]
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


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. AddisonWesley, 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

A000040 complement A002313.


MAPLE

A002145 := proc(n)
option remember;
if n = 1 then
3;
else
a := nextprime(procname(n1)) ;
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 *)


PROG

(PARI) forprime(p=2, 1e3, if(p%4==3, print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011
(Haskell)
a002145 n = a002145_list !! (n1)
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


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



