%I M2624 N1039 #299 Sep 03 2024 00:58:20
%S 3,7,11,19,23,31,43,47,59,67,71,79,83,103,107,127,131,139,151,163,167,
%T 179,191,199,211,223,227,239,251,263,271,283,307,311,331,347,359,367,
%U 379,383,419,431,439,443,463,467,479,487,491,499,503,523,547,563,571
%N Primes of the form 4*k + 3.
%C 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
%C Primes which are not the sum of two squares, see the comment in A022544. - _Artur Jasinski_, Nov 15 2006
%C Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)
%C Inert rational primes in the field Q(sqrt(-1)). - _N. J. A. Sloane_, Dec 25 2017
%C Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1. - _Benoit Cloitre_, Oct 22 2002
%C 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
%C 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
%C Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - _Alexander Adamchuk_, Nov 30 2006
%C Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - _Alexander Adamchuk_, Apr 18 2007
%C This sequence is a proper subset of the set of the absolute values of negative fundamental discriminants (A003657). - _Paul Muljadi_, Mar 29 2008
%C Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after _Paul Curtz_, Sep 10 2008
%C A079261(a(n)) = 1; complement of A145395. - _Reinhard Zumkeller_, Oct 12 2008
%C Subsequence of A007970. - _Reinhard Zumkeller_, Jun 18 2011
%C A151763(a(n)) = -1.
%C 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)
%C 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
%C Numbers n > 2 such that ((n-2)!!)^2 == 1 (mod n). - _Thomas Ordowski_, Jul 24 2016
%C Odd numbers n > 1 such that ((n-1)!!)^2 == 1 (mod n). - _Thomas Ordowski_, Jul 25 2016
%C Primes p such that (p-2)!! == (p-3)!! (mod p). - _Thomas Ordowski_, Jul 28 2016
%C See Granville and Martin for a discussion of the relative numbers of primes of the form 4k+1 and 4k+3. - Editors, May 01 2017
%C Sometimes referred to as Blum primes for their connection to A016105 and the Blum Blum Shub generator. - _Charles R Greathouse IV_, Jun 14 2018
%C Conjecture: a(n) for n > 4 can be written as a sum of 3 primes of the form 4k+1, which would imply that primes of the form 4k+3 >= 23 can be decomposed into a sum of 6 nonzero squares. - _Thomas Scheuerle_, Feb 09 2023
%D 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 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 W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 66.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.
%H Zak Seidov, <a href="/A002145/b002145.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H M. Abramowitz and I. A. Stegun, eds., <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H D. Alpern, <a href="https://www.alpertron.com.ar/GAUSSPR.HTM">Gaussian primes</a>.
%H Lenore Blum, Manuel Blum, and Mike Shub, <a href="https://pdfs.semanticscholar.org/c19b/91cdc1da67c52e606cd4752472ce0db83131.pdf">A simple unpredictable pseudo-random number generator</a>, SIAM Journal on Computing 15:2 (1 May 1986), pp. 364-383.
%H A. Granville and G. Martin, <a href="https://arxiv.org/abs/math/0408319">Prime number races</a>, arXiv:math/0408319 [math.NT], 2004.
%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, <a href="https://arxiv.org/abs/1802.08349">On a Dynamical Approach to Some Prime Number Sequences</a>, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
%H E. T. Ordman, <a href="/A002143/a002143.pdf">Tables of the class number for negative prime discriminants</a>, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]
%H H. J. Smith, <a href="https://web.archive.org/web/20121224064852/http://harry-j-smith-memorial.com:80/GPrimes/index.html">Gaussian Primes</a>.
%H I. Stewart, <a href="https://books.google.ru/books?id=I-RSVN6TjXsC&printsec=frontcover&dq=%22Greatest%22+ian+stewart&hl=en&sa=X&redir_esc=y#v=onepage&q=Riemann&f=false">The Great Mathematical Problems</a>, 2013.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian Prime</a>.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianInteger.html">Gaussian Integer</a>.
%H Wolfram Research, <a href="https://functions.wolfram.com/NumberTheoryFunctions/JacobiSymbol/31/01/ShowAll.html">The Gauss Reciprocity Law</a>.
%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>.
%H <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>.
%F Remove from A000040 terms that are in A002313.
%F Intersection of A000040 and A004767. - _Alonso del Arte_, Apr 22 2014
%F From _Vaclav Kotesovec_, Apr 30 2020: (Start)
%F Product_{k>=1} (1 - 1/a(k)^2) = A243379.
%F Product_{k>=1} (1 + 1/a(k)^2) = A243381.
%F Product_{k>=1} (1 - 1/a(k)^3) = A334427.
%F Product_{k>=1} (1 + 1/a(k)^3) = A334426.
%F Product_{k>=1} (1 - 1/a(k)^4) = A334448.
%F Product_{k>=1} (1 + 1/a(k)^4) = A334447.
%F Product_{k>=1} (1 - 1/a(k)^5) = A334452.
%F Product_{k>=1} (1 + 1/a(k)^5) = A334451. (End)
%F From _Vaclav Kotesovec_, May 05 2020: (Start)
%F Product_{k>=1} (1 + 1/a(k)) / (1 + 1/A002144(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
%F Product_{k>=1} (1 - 1/a(k)) / (1 - 1/A002144(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
%F Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log(2 * (2^(n*s) - 1) * (n*s - 1)! * zeta(n*s) / (Pi^(n*s) * abs(EulerE(n*s - 1))))/n, s >= 3 odd number. - _Dimitris Valianatos_, May 20 2020
%p A002145 := proc(n)
%p option remember;
%p if n = 1 then
%p 3;
%p else
%p a := nextprime(procname(n-1)) ;
%p while a mod 4 <> 3 do
%p a := nextprime(a) ;
%p end do;
%p return a;
%p end if;
%p end proc:
%p seq(A002145(n),n=1..20) ; # _R. J. Mathar_, Dec 08 2011
%t Select[4Range[150] - 1, PrimeQ] (* _Alonso del Arte_, Dec 19 2013 *)
%t Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] == 1 &] (* or *)
%t Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == -1 &] (* _Robert G. Wilson v_, May 11 2014 *)
%o (PARI) forprime(p=2,1e3,if(p%4==3,print1(p", "))) \\ _Charles R Greathouse IV_, Jun 10 2011
%o (Haskell)
%o a002145 n = a002145_list !! (n-1)
%o a002145_list = filter ((== 1) . a010051) [3, 7 ..]
%o -- _Reinhard Zumkeller_, Aug 02 2015, Sep 23 2011
%o (Magma) [4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // _Arkadiusz Wesolowski_, Nov 15 2013
%o (Sage)
%o def A002145_list(n): return [p for p in prime_range(1, n + 1) if p % 4 == 3] # _Peter Luschny_, Jul 29 2014
%Y Cf. A000032, A002144, A003657, A085992, A122869, A122870, A334912.
%Y Cf. A000408, A005098, A095278, A016754.
%Y Apart from initial term, same as A045326.
%Y Cf. A016105.
%Y Cf. A004614 (multiplicative closure).
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Apr 21 2000