Search: id:A002145
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%I A002145 M2624 N1039
%S A002145 3,7,11,19,23,31,43,47,59,67,71,79,83,103,107,127,131,139,151,163,167,
%T A002145 179,191,199,211,223,227,239,251,263,271,283,307,311,331,347,359,367,
%U A002145 379,383,419,431,439,443,463,467,479,487,491,499,503,523,547,563,571
%N A002145 Primes of form 4n+3.
%C A002145 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 (njas(AT)research.att.com), 
               Jun 28 2008
%C A002145 Primes which are not the sum of two squares. - Artur Jasinski (grafix(AT)csl.pl), 
               Nov 15 2006
%C A002145 Natural primes which are also Gaussian primes. (It is a common error 
               to refer to this sequence as "the Gaussian primes".)
%C A002145 sin(a(n)*pi/2) = -1 with pi=3.1415..., see A070750. - Reinhard Zumkeller 
               (reinhard.zumkeller(AT)gmail.com), May 04 2002
%C A002145 Numbers n such that the product of coefficients of (2n)-th cyclotomic 
               polynomial equals -1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Oct 22 2002
%C A002145 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 (blekraj(AT)yahoo.com), Jul 17 2003
%C A002145 Also primes p that divide Lucas[(p-1)/2] or Lucas[(p+1)/2], where Lucas[n] 
               = A000032[n]. Union of A122869 and A122870. - Alexander Adamchuk 
               (alex(AT)kolmogorov.com), Sep 16 2006
%C A002145 Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Nov 30 2006
%C A002145 Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - Alexander 
               Adamchuk (alex(AT)kolmogorov.com), Apr 18 2007
%C A002145 This sequence is a proper subset of the set of the absolute values of 
               negative fundamental discriminants (A003657). - Paul Muljadi (paulmuljadi(AT)yahoo.com), 
               Mar 29 2008
%C A002145 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 (bpcrtz(AT)free.fr), 
               Sep 10 2008]
%C A002145 A079261(a(n)) = 1; complement of A145395. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Oct 12 2008]
%C A002145 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 (c.zizka(AT)email.cz), Oct 20 2009]
%D A002145 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.
%D A002145 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, p. 219, th. 252.
%D A002145 W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 
               2 vols., 1956, Vol. 1, p. 66.
%D A002145 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002145 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A002145 T. D. Noe, <a href="b002145.txt">Table of n, a(n) for n=1..1000</a>
%H A002145 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A002145 D. Alpern, <a href="http://www.alpertron.com.ar/GAUSSPR.HTM">Gaussian 
               primes</a>
%H A002145 A. Granville and G. Martin, <a href="http://www.arXiv.org/abs/math.NT/
               0408319">Prime number races</a>
%H A002145 H. J. Smith, <a href="http://harry-j-smith.com/hjsmithh/GPrimes/index.html">
               Gaussian Primes</a>
%H A002145 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GaussianPrime.html">Link to a section of The World of Mathematics.</
               a>
%H A002145 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               GaussianInteger.html">"Gaussian Integer"</a>.
%H A002145 Wolfram Research, <a href="http://functions.wolfram.com/NumberTheoryFunctions/
               JacobiSymbol/31/01/ShowAll.html">The Gauss Reciprocity Law</a>
%H A002145 <a href="Sindx_Ga.html#gaussians">Index entries for Gaussian integers 
               and primes</a>
%t A002145 lst={};Do[If[PrimeQ[p=4*n+3], (*Print[p];*)AppendTo[lst, p]], {n, 0, 
               9^2}];lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Aug 21 
               2008]
%Y A002145 Cf. A002144. Apart from initial term, same as A045326.
%Y A002145 Cf. A122869, A122870, A000032.
%Y A002145 A000040 \setminus A002313
%Y A002145 Cf. A003657.
%Y A002145 Sequence in context: A131426 A080978 A160216 this_sequence A092109 A117991 
               A118260
%Y A002145 Adjacent sequences: A002142 A002143 A002144 this_sequence A002146 A002147 
               A002148
%K A002145 nonn,easy
%O A002145 1,1
%A A002145 N. J. A. Sloane (njas(AT)research.att.com).
%E A002145 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 21 2000

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