A128107 -id:A128107 - cp's OEIS frontend

A128106 Sizes of possible gaps around a Gaussian prime: 1 and the even numbers in A001481.

1, 2, 4, 8, 10, 16, 18, 20, 26, 32, 34, 36, 40, 50, 52, 58, 64, 68, 72, 74, 80, 82, 90, 98, 100, 104, 106, 116, 122, 128, 130, 136, 144, 146, 148, 160, 162, 164, 170, 178, 180, 194, 196, 200, 202, 208, 212, 218, 226, 232, 234, 242, 244, 250, 256, 260, 272, 274

Offset: 1

T. D. Noe, Feb 15 2007

nonn

For a given Gaussian prime u, the size of its gap is the minimum of norm(u-v) as v varies over all other Gaussian primes, where norm(a+b*i)=a^2+b^2. Only the small Gaussian primes 1+i and 2+i (and their associates and reflections) have gaps of diameter 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A128107, A128108, A128109.

q=12; imax=2*q^2; lst=Select[Union[Flatten[Table[2*x^2+2*y^2, {x,0,q}, {y,0,x}]]], #<=imax&]; Join[{1},Drop[lst,1]] (* Vladimir Joseph Stephan Orlovsky, Apr 20 2011 *)

def A128106_list(max):
    R = []; s = 1; sq = 1
    for n in (0..max//2):
        if n == s:
            sq += 1;
            s = sq*sq;
        for k in range(sq):
            if is_square(n-k*k):
                R.append(2*n)
                break
    R[0] = 1
    return R
A128106_list(274) # Peter Luschny, Jun 20 2014

A128108 Imaginary part of the smallest Gaussian prime having a gap size of exactly A128106(n).

Original entry on oeis.org

1, 0, 12, 8, 5, 0, 48, 24, 320, 281, 0, 1688, 632, 2951, 3914, 2752, 0

Offset: 1

T. D. Noe, Feb 15 2007

nonn

Due to symmetry, there are actually 4 or 8 Gaussian primes having this gap size. We list the prime in the first octant of the complex plane.

Cf. A128107, A128109.

A128109 Smallest positive real Gaussian prime having a gap size of exactly A128106(n).

Original entry on oeis.org

3, 47, 31, 107, 79, 2447, 523, 1019, 11311, 563, 13411, 7559, 24943, 46867, 72103, 14243, 35759, 110339, 112967, 106591, 50023, 275323, 69991, 189251, 1267907, 849143, 1446719, 2382979, 3922691, 3166679, 6130459, 959207, 4100479, 10134671

Offset: 2

T. D. Noe, Feb 15 2007

nonn

These Gaussian primes have the form p+0*i, where p is an ordinary (rational) prime with p=3 (mod 4). Gethner et al. consider gaps around real Gaussian primes. In general, the Gaussian primes whose real and imaginary parts are given in A128107 and A128108 have a smaller magnitude than the real Gaussian primes given here. However, these real Gaussian primes are easier to compute.

Ellen Gethner, Stan Wagon and Brian Wick, A stroll through the Gaussian primes, American Mathematical Monthly 105 (1998), 327-337.

cp's OEIS Frontend

A128106 Sizes of possible gaps around a Gaussian prime: 1 and the even numbers in A001481.

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A128108 Imaginary part of the smallest Gaussian prime having a gap size of exactly A128106(n).

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Author

Keywords

Comments

Crossrefs

A128109 Smallest positive real Gaussian prime having a gap size of exactly A128106(n).

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Author

Keywords

Comments

Links