1,1

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

Primes which are not the sum of two squares. - Artur Jasinski (grafix(AT)csl.pl), 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 (reinhard.zumkeller(AT)gmail.com), May 04 2002

Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1 - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (blekraj(AT)yahoo.com), Jul 17 2003

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

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

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

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

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]

A079261(a(n)) = 1; complement of A145395. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 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 (c.zizka(AT)email.cz), Oct 20 2009]

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).

T. D. Noe, Table of n, a(n) for n=1..1000

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, Link to a section of The World of Mathematics.

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

Wolfram Research, The Gauss Reciprocity Law

Index entries for Gaussian integers and primes

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]

Cf. A002144. Apart from initial term, same as A045326.

Cf. A122869, A122870, A000032.

A000040 \setminus A002313

Cf. A003657.

Sequence in context: A131426 A080978 A160216 this_sequence A092109 A117991 A118260

Adjacent sequences: A002142 A002143 A002144 this_sequence A002146 A002147 A002148

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 21 2000