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 A002145 Primes of the form 4*k + 3. (Formerly M2624 N1039) 314
 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".) Inert rational primes in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017 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 hypotenuse of 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 Numbers n > 2 such that ((n-2)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 24 2016 Odd numbers n > 1 such that ((n-1)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 25 2016 Primes p such that (p-2)!! == (p-3)!! (mod p). - Thomas Ordowski, Jul 28 2016 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 Sometimes referred to as Blum primes for their connection to A016105 and the Blum Blum Shub generator. - Charles R Greathouse IV, Jun 14 2018 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 Zak Seidov, 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 Lenore Blum, Manuel Blum, and Mike Shub, A simple unpredictable pseudo-random number generator, SIAM Journal on Computing 15:2 (1 May 1986), pp. 364-383. A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004. Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33. Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018. E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes] H. J. Smith, Gaussian Primes I. Stewart, The Great Mathematical Problems, 2013. Eric Weisstein's World of Mathematics, Gaussian Prime Eric Weisstein's World of Mathematics, "Gaussian Integer". Wolfram Research, The Gauss Reciprocity Law FORMULA Remove from A000040 terms that are in A002313. Intersection of A000040 and A004767. - Alonso del Arte, Apr 22 2014 From Vaclav Kotesovec, Apr 30 2020: (Start) Product_{k>=1} (1 - 1/a(k)^2) = A243379. Product_{k>=1} (1 + 1/a(k)^2) = A243381. Product_{k>=1} (1 - 1/a(k)^3) = A334427. Product_{k>=1} (1 + 1/a(k)^3) = A334426. Product_{k>=1} (1 - 1/a(k)^4) = A334448. Product_{k>=1} (1 + 1/a(k)^4) = A334447. Product_{k>=1} (1 - 1/a(k)^5) = A334452. Product_{k>=1} (1 + 1/a(k)^5) = A334451. (End) From Vaclav Kotesovec, May 05 2020: (Start) Product_{k>=1} (1 + 1/a(k)) / (1 + 1/A002144(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962... Product_{k>=1} (1 - 1/a(k)) / (1 - 1/A002144(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End) 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 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 - 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 ((== 1) . a010051) [3, 7 ..] -- Reinhard Zumkeller, Aug 02 2015, Sep 23 2011 (Magma) [4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // Arkadiusz Wesolowski, Nov 15 2013 (Sage) def A002145_list(n): return [p for p in prime_range(1, n + 1) if p % 4 == 3]  # Peter Luschny, Jul 29 2014 CROSSREFS Cf. A000032, A002144, A003657, A085992, A122869, A122870, A334912. Apart from initial term, same as A045326. Cf. A016105. Cf. A004614 (multiplicative closure). Sequence in context: A160216 A181516 A285015 * A002052 A092109 A277878 Adjacent sequences:  A002142 A002143 A002144 * A002146 A002147 A002148 KEYWORD nonn,easy,changed AUTHOR EXTENSIONS More terms from James A. Sellers, Apr 21 2000 STATUS approved

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Last modified October 2 19:14 EDT 2022. Contains 357228 sequences. (Running on oeis4.)