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A355431
Numbers k whose binary expansion, when interpreted in base -1+i, gives a Gaussian prime.
1
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
1,1
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
LINKS
John-Vincent Saddic, Graphs on the complex plane
John-Vincent Saddic, Julia program
John-Vincent Saddic, Python program
EXAMPLE
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.
PROG
(Julia) # See links.
(Python) # See links.
CROSSREFS
Cf. A066321 (real integers in base -1+i).
Sequence in context: A342733 A138970 A168550 * A244737 A285347 A187836
KEYWORD
nonn,base
AUTHOR
John-Vincent Saddic, Jul 17 2022
STATUS
approved