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These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).

With the restriction Re(z) >= Im(z) >= 0 used here and in A239621, there is exactly one Gaussian prime z for each possible norm |z|^2 in A055025. Sequence A239397 lists both, (x, y) and (y, x), for each of these having x > y (i.e., except for x = y = 1).

According to the graph, the values seem rather uniformly distributed between 0 and the upper bound sqrt(A055025(n)/2) ~ sqrt(n log n), in contrast to the values of the real parts A300587(n).

We walk through the Gaussian primes in the first octant of the Gaussian plane along increasing norm: 1+i, 2+i, 3, 3+2i, 4+i, 5+2i, 6+i, 5+4i, 7, 7+2i etc. The sequence lists the squared distance between consecutive Gaussian primes along this walk.

Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.

Closed under multiplication. - David W. Wilson, Dec 20 2004

Also, numbers whose cubes are the sum of 2 squares. - Artur Jasinski, Nov 21 2006 (Cf. A125110.)

Terms are the squares of smallest radii of circles covering (on a square grid) a number of points equal to the terms of A057961. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 16 2007. [Comment corrected by T. D. Noe, Mar 28 2008]

Numbers with more 4k+1 divisors than 4k+3 divisors. If a(n) is a member of this sequence, then so too is any power of a(n). - Ant King, Oct 05 2010

A000161(a(n)) > 0; A070176(a(n)) = 0. - Reinhard Zumkeller, Feb 04 2012, Aug 16 2011

Numbers that are the norms of Gaussian integers. This sequence has unique factorization; the primitive elements are A055025. - Franklin T. Adams-Watters, Nov 25 2011

These are numbers n such that all of n's odd prime factors congruent to 3 modulo 4 occur to an even exponent (Fermat's two-squares theorem). - Jean-Christophe Hervé, May 01 2013

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. The present sequence without 0 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. Then A055025 (norms of Gaussian primes) gives the "prime divisors" of the square lattice. - Jean-Christophe Hervé, May 01 2013

For any i,j > 0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. - Boris Putievskiy, May 05 2013

The sequence is closed under multiplication. Primitive elements are in A055025. The sequence can be split into 3 multiplicatively closed subsequences: {0}, A004431 and A125853. - Jean-Christophe Hervé, Nov 17 2013

Generalizing Jasinski's comment, same as numbers whose odd powers are the sum of 2 squares, by Fermat's two-squares theorem. - Jonathan Sondow, Jan 24 2014

By the 4 squares theorem, every nonnegative integer can be expressed as the sum of two elements of this sequence. - Franklin T. Adams-Watters, Mar 28 2015

There are never more than 3 consecutive terms. Runs of 3 terms start at 0, 8, 16, 72, ... (A082982). - Ivan Neretin, Nov 09 2015

Conjecture: barring the 0+2, 0+4, 0+8, 0+16, ... sequence, the sum of 2 distinct terms in this sequence is never a power of 2. - J. Lowell, Jan 14 2022

All the areas of squares whose vertices have integer coordinates. - Neeme Vaino, Jun 14 2023

Numbers represented by the definite binary quadratic forms x^2 + 2nxy + (n^2+1)y^2 for any integer n. This sequence contains the even powers of any integer. An odd power of a number appears only if the number itself belongs to the sequence. The equation given in the comment by Boris Putievskiy 2013 is Brahmagupta's identity with n = 1. It proves that any set of numbers of the form a^2 + nb^2 is closed under multiplication. - Klaus Purath, Sep 06 2023

These are the norms of the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Let us say that an integer n divides a lattice if there exists a sublattice of index n. Example: 3 divides the hexagonal lattice. Then A003136 (Loeschian numbers) is the sequence of divisors of the hexagonal 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 hexagonal lattice. Similarly, A055025 (Norms of Gaussian primes) is the sequence of "prime divisors" of the square lattice. - Jean-Christophe Hervé, Dec 04 2006

Definition of Gaussian primes (Pieper, Die komplexen Zahlen, p. 122): 1) i+i, norm N(i+i) = 2. 2) Natural primes p with p = 3 mod 4, norm N(p) = p^2. 3) primes a+bi, a>0, b>0 with a^2 + b^2 = p = 1 mod 4, p natural prime. Norm N(a+bi) = p. b+ai is a different Gaussian prime number, b+ai cannot be factored into a+bi and a unit. 4) All complex numbers from 1) to 3) multiplied by the units -1,i,-i, these are the associated numbers. The sequence contains all the Gaussian primes mentioned in 1) - 3).

Every complex number can be factored completely into the Gaussian prime numbers defined by the sequence, an additional unit as factor can be necessary. This factorization can be used to calculate the complex sigma, as defined by Spira. The elements a(n) are ordered by the size of their norm. If the two different primes a+bi and b+ai have the same norm, they are ordered by the size of the real part of the complex prime number. So a+bi follows b+ai in the sequence, if a > b.

Of course this is not the only possible definition. As primes p = 1 mod 4 can be factored in p = (-i)(a+bi)(b+ai) and the norm N(a+bi) = N(b+ai) = p, these primes a+bi occur much earlier in the sequence than p does in the sequence of natural primes. 4+5i with norm 41 occurs before prime 7 with norm 49.

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-i).

Consists of primes congruent to 1, 2, 3, 5, 7, 9 (mod 20) together with the squares of all other primes.

From Jianing Song, Feb 20 2021: (Start)

The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.

Note that Z[sqrt(-5)] has class number 2.

For primes p == 1, 9 (mod 20), there are two distinct ideals with norm p in Z[sqrt(-5)], namely (x + y*sqrt(-5)) and (x - y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p.

For p == 3, 7 (mod 20), there are also two distinct ideals with norm p, namely (p, x+y*sqrt(-5)) and (p, x-y*sqrt(-5)), where (x,y) is a solution to x^2 + 5*y^2 = p^2 with y != 0; (2, 1+sqrt(-5)) and (sqrt(-5)) are respectively the unique ideal with norm 2 and 5.

For p == 11, 13, 17, 19 (mod 20), (p) is the only ideal with norm p^2. (End)

Also norms of prime ideals in Z[sqrt(-2)], which is a unique factorization domain. The norm of a nonzero ideal I in a ring R is defined as the size of the quotient ring R/I.

Consists of the primes congruent to 1, 2, 3 modulo 8 and the squares of primes congruent to 5, 7 modulo 8.

For primes p == 1, 3 (mod 8), there are two distinct ideals with norm p in Z[sqrt(2)], namely (x + y*sqrt(-2)) and (x - y*sqrt(-2)), where (x,y) is a solution to x^2 + 2*y^2 = p; for p = 2, (sqrt(-2)) is the unique ideal with norm p; for p == 5, 7 (mod 8), (p) is the only ideal with norm p^2.