A355431 Numbers k whose binary expansion, when interpreted in base -1+i, gives a Gaussian prime.

2, 5, 6, 9, 11, 13, 14, 15, 17, 19, 21, 23, 25, 27, 31, 33, 37, 39, 41, 43, 49, 51, 53, 57, 58, 59, 63, 69, 71, 73, 77, 81, 83, 89, 97, 99, 101, 111, 113, 117, 119, 123, 127, 129, 131, 133, 137, 139, 141, 147, 159, 163, 169, 177, 183, 191, 193, 197, 201, 207

Offset: 1

Author

John-Vincent Saddic, Jul 17 2022

Keywords

• nonn
• base

Comments

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

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

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.

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.

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

Examples

			123 is a term since z(123) = 2+7i is a Gaussian prime.
124 is not a term because z(124) = 2+4i is not a Gaussian prime.
		

Links

• John-Vincent Saddic, Graphs on the complex plane
• John-Vincent Saddic, Julia program
• John-Vincent Saddic, Python program

Crossrefs

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

Programs

Julia

# See links.

Python

# See links.