1,1

These are the primitive elements of A009003.

-1 is a quadratic residue mod a prime p iff p is in this sequence.

sin(a(n)*pi/2) = 1 with pi=3.1415..., see A070750. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 04 2002

If at least one of the odd primes p, q belongs to the sequence, then either both or neither of the congruences x^2=p (mod q), x^2=q (mod p) are solvable, according to Gauss reciprocity law. - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 17 2003

Odd primes such that binomial(p-1,(p-1)/2) == 1 (mod p) - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 07 2004

Primes that are the hypotenuse of a right triangle with integer sides. The Pythagorean triple is {A002365(n+4), A002366(n+4),a(n)}.

Also, primes of the form a^k + b^k, k >1 (cf. A089716). - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 17 2003

The square of A002144(n) is the average of two other squares. This fact gives rise to a class of monic polynomials x^2 + bx + c with b = A002144(n) that will factor over the integers regardless of the sign of c. See A114200. - Owen Mertens (owenmertens(AT)missouristate.edu), Nov 16 2005

Also such primes p that the last digit is always 1 for the Nexus numbers of form n^p - (n-1)^p. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 10 2006

The set of Pythagorean primes is a proper subset of the set of positive fundamental discriminants (A003658). - Paul Muljadi (paulmuljadi(AT)yahoo.com), Mar 28 2008

Frenicle mentionned 4n+1 for primes : Methode pour trouver .., page 14 on 44.In Divers ouvrages de mathematique .. .In-folio,6,518,1 pp,Paris,1693. [From Paul Curtz (bpcrtz(AT)free.fr), Sep 05 2008]

A079260(a(n)) = 1; complement of A137409. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 11 2008]

Contribution from Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008: (Start)

If we take 4 numbers : 1, A002314(n), A152676(n), A152680(n) then

multiplication table modulo A002144(n) is isomorphc to the Latin square:

1 2 3 4

2 4 1 3

3 1 4 2

4 3 2 1

and isomorphic to the multiplication table of {1, I, -I, -1} where I is Sqrt[ -1],

A152680(n) is isomorphic to -1, A002314(n) with I or -I and A152676(n) vice versa -I or I.

1, A002314(n), A152676(n), A152680(n) are subfield of Galois Field [A002144(n)]. (End)

Primes p such that arithmetic mean of divisors of p^3 is an integer. There are 2 sequences of such primes, this one and A002145. [From Ctibor O. Zizka (c.zizka(AT)email.cz), Oct 20 2009]

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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. Cox, "Primes of Form x^2 + n y^2", Wiley, 1989.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 76.

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. 70 (4) (1997) 263.

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].

C. Banderier, Calcul de (-1/p)

J. Butcher, The Quadratic Residue Theorem

R. Chapman, Quadratic reciprocity

J. E. Ewell, A Simple Proof of Fermat's Two-Square Theorem

A. Granville and G. Martin, Prime number races

D. & C. Hazzlewood, Quadratic Reciprocity

R. C. Laubenbacher & D. J. Pengelley, Eisenstein's Misunderstood Geometric Proof of the Quaratic Reciprocity Theorem

R. C. Laubenbacher & D. J. Pengelley, Gauss, Eisenstein and the -third' proof of the Quadratic Reciprocity Theorem

K. Matthews, Serret's algorithm based Server

Eric Weisstein's World of Mathematics, Wilson's Theorem

Eric Weisstein's World of Mathematics, Pythagorean Triples

Wolfram Research, The Gauss Reciprocity Law

G. Xiao, Two squares

Wikipedia, Quadratic reciprocity

Odd primes of form x^2 + y^2, (x=A002331, y=A002330, with x<y) or of form u^2 + 4*v^2, (u=A002972, v=A002973, with u odd). - Lekraj Beedassy (blekraj(AT)yahoo.com), Jul 16 2004

p^2-1=12*sum_{i=0..floor(p/4)} floor[sqrt(i*p)] where p=a(n)=4n+1 [Shirali].

The following table shows the relationship

between several closely related sequences:

Here p = A002144 = primes == 1 mod 4, p = a^2+b^2 with a < b;

a = A002331, b = A002330, t_1 = ab/2 = A070151;

p^2 = c^2+d^2 with c < d; c = A002366, d = A002365,

t_2 = 2ab = A145046, t_3 = b^2-a^2 = A070079,

with {c,d} = {t_2, t_3}, t_4 = cd/2 = ab(b^2-a^2).

---------------------------------

.p..a..b..t_1..c...d.t_2.t_3..t_4

---------------------------------

.5..1..2...1...3...4...4...3....6

13..2..3...3...5..12..12...5...30

17..1..4...2...8..15...8..15...60

29..2..5...5..20..21..20..21..210

37..1..6...3..12..35..12..35..210

41..4..5..10...9..40..40...9..180

53..2..7...7..28..45..28..45..630

.................................

a := []; for n from 1 to 500 do if isprime(4*n+1) then a := [op(a), 4*n+1]; fi; od: A002144 := n->a[n];

Select[4*Range[140] + 1, PrimeQ[ # ] &] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 16 2006

aa = {}; Do[If[Mod[Prime[n], 4] == 1, AppendTo[aa, Prime[n]]], {n, 1, 200}]; aa [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]

lst={}; Do[Do[p=n^2+m^2; If[PrimeQ[p], AppendTo[lst, p]], {n, 0, 5!}], {m, 0, 5!}]; lst; Take[Union[lst], 123] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 21 2009]

For values of n see A005098. Cf. A002145, A002476. Apart from initial term, same as A002313.

Cf. A114200.

Cf. A003658.

A002314, A152676, A152680. [From Artur Jasinski (grafix(AT)csl.pl), Dec 10 2008]

Adjacent sequences: A002141 A002142 A002143 this_sequence A002145 A002146 A002147

Sequence in context: A078900 A113482 A077426 this_sequence A111055 A145016 A123079

nonn,easy,nice,new

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

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