cp's OEIS Frontend

A055025 Norms of Gaussian primes.

%I A055025 #70 Feb 16 2025 08:32:42
%S A055025 2,5,9,13,17,29,37,41,49,53,61,73,89,97,101,109,113,121,137,149,157,
%T A055025 173,181,193,197,229,233,241,257,269,277,281,293,313,317,337,349,353,
%U A055025 361,373,389,397,401,409,421,433,449,457,461,509,521,529,541,557,569
%N A055025 Norms of Gaussian primes.
%C A055025 This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - _Wolfdieter Lang_, Mar 24 2014, edited by _M. F. Hasler_, Mar 09 2018
%C A055025 From _Jean-Christophe Hervé_, May 01 2013: (Start)
%C A055025 The present sequence is related to the square lattice, and to its division in square sublattices. Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. Then A001481 (norms of Gaussian integers) is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. The present sequence gives the "prime divisors" of the square lattice.
%C A055025 Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)
%C A055025 The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - _Jean-Christophe Hervé_, Nov 17 2013
%D A055025 R. K. Guy, Unsolved Problems in Number Theory, A16.
%D A055025 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
%H A055025 Amiram Eldar, <a href="/A055025/b055025.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A055025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianPrime.html">Gaussian prime</a>
%H A055025 Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_Gaussian_integer_factorizations">Table of Gaussian integer factorizations</a>
%H A055025 <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%F A055025 Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].
%F A055025 a(n) ~ 2n log n. - _Charles R Greathouse IV_, Feb 06 2017
%e A055025 There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i). In A239621 2+i is listed as 2, 1.
%t A055025 Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* _Jean-François Alcover_, Apr 08 2011 *)
%t A055025 (* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3]  (* _Jean-François Alcover_, Dec 07 2012 *)
%o A055025 (PARI) list(lim)=my(v=List()); if(lim>=2, listput(v,2)); forprime(p=3,sqrtint(lim\1), if(p%4==3, listput(v,p^2))); forprime(p=5,lim, if(p%4==1, listput(v,p))); Set(v) \\ _Charles R Greathouse IV_, Feb 06 2017
%o A055025 (PARI) isA055025(n)=(isprime(n) && n%4<3) || (issquare(n, &n) && isprime(n) && n%4==3) \\ _Jianing Song_, Aug 15 2023, based on _Charles R Greathouse IV_'s program for A055664
%Y A055025 Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.
%Y A055025 Cf. A239397, A239621 (Gaussian primes).
%K A055025 nonn,easy,nice
%O A055025 1,1
%A A055025 _N. J. A. Sloane_, Jun 09 2000
%E A055025 More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000