A355431 - OEIS

%I #47 Mar 31 2024 12:05:08

%S 2,5,6,9,11,13,14,15,17,19,21,23,25,27,31,33,37,39,41,43,49,51,53,57,

%T 58,59,63,69,71,73,77,81,83,89,97,99,101,111,113,117,119,123,127,129,

%U 131,133,137,139,141,147,159,163,169,177,183,191,193,197,201,207

%N Numbers k whose binary expansion, when interpreted in base -1+i, gives a Gaussian prime.

%C Complex base -1+i is a bijection between integers k and Gaussian integers z(k) = A318438(k) + A318439(k)*i.

%C The present sequence is those k where z(k) is a Gaussian prime.

%C The Gaussian primes have an 8-way symmetry in the complex plane so that this sequence is also the Gaussian primes in the conjugate complex base -1-i.

%C The graphs on the complex plane (see links) show the Gaussian primes mapped and connected by lines in the order in which their indices appear in {a(n)}. The numbers in base -1+i tile the complex plane in the twin dragon fractal pattern, and the Gaussian primes are numerous such that the fractal is still discernible.

%C The only even terms are 2, 6, 14, and 58, since even terms correspond to Gaussian integers divisible by -1+i, and the base-(-1+i) expansions of -1+i, -1-i, 1+i, and 1-i are 10, 110, 1110, and 111010 respectively. - _Jianing Song_, Oct 02 2022

%H John-Vincent Saddic, <a href="/A355431/a355431.png">Graphs on the complex plane</a>

%H John-Vincent Saddic, <a href="/A355431/a355431.jl.txt">Julia program</a>

%H John-Vincent Saddic, <a href="/A355431/a355431_3.py.txt">Python program</a>

%e 123 is a term since z(123) = 2+7i is a Gaussian prime.

%e 124 is not a term because z(124) = 2+4i is not a Gaussian prime.

%o (Julia) # See links.

%o (Python) # See links.

%Y Cf. A066321 (real integers in base -1+i).

%K nonn,base

%O 1,1

%A _John-Vincent Saddic_, Jul 17 2022