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A262433 Quater-imaginary representation of the Gaussian primes with an even imaginary part. 0
3, 11, 13, 21, 31, 101, 111, 113, 123, 133, 201, 211, 213, 223, 233, 301, 321, 323, 331, 1003, 1011, 1021, 1031, 1033, 1101, 1113, 1123, 1131, 1133, 1201, 1203, 1213, 1223, 1231, 1233, 1311, 1321, 1323, 2001, 2011, 2031, 2033, 2103, 2113, 2131, 2133, 2203 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Not all Gaussian primes will be in this list as complex numbers with an odd imaginary part require a value after the radix point (".") in the quater-imaginary number system.

LINKS

Table of n, a(n) for n=1..47.

Donald Knuth, An imaginary number system, Communications of the ACM 3 (4), April 1960, pp. 245-247.

OEIS Wiki, Quater-imaginary base

OEIS Wiki, Gaussian primes

Wikipedia, Quater-imaginary base

EXAMPLE

1231_(2i) = 1(2i)^3 + 2(2i)^2 + 3(2i)^1 + 1(2i)^0 = -7-2i which is a Gaussian prime.

PROG

(C++)

#include <stdlib.h>

#include <math.h>

#include <iostream>

#include <fstream>

using namespace std;

bool isPrime(int n)

{

  if (n <= 1)

    return false;

  if (n == 2)

    return true;

  if (n % 2 == 0)

    return false;

  int rootNCeil = (int)sqrt(n) + 1;

  for (int div = 3; div <= rootNCeil; div+=2)

  {

    if (n % div == 0)

      return false;

  }

  return true;

}

int main()

{

  const int maxDigits = 8;

  unsigned int maxVal = 2 << ((maxDigits * 2) - 1);

  for (unsigned int n = 0; n < maxVal; n++)

  {

    // Split binary representation of n into real part and imaginary part

    int rp = (n & 0x00000003) - ((n & 0x00000030) >> 2) +

      ((n & 0x00000300) >> 4) - ((n & 0x00003000) >> 6) +

      ((n & 0x00030000) >> 8) - ((n & 0x00300000) >> 10) +

      ((n & 0x03000000) >> 12) - ((n & 0x30000000) >> 14);

    int ip = ((n & 0x0000000c) >> 1) - ((n & 0x000000c0) >> 3) +

      ((n & 0x00000c00) >> 5) - ((n & 0x0000c000) >> 7) +

      ((n & 0x000c0000) >> 9) - ((n & 0x00c00000) >> 11) +

      ((n & 0x0c000000) >> 13) - ((n & 0xc0000000) >> 15);

    if ((ip == 0 && (abs(rp) % 4 == 3) && isPrime(abs(rp))) ||

        (rp == 0 && (abs(ip) % 4 == 3) && isPrime(abs(ip))) ||

        (rp != 0 && ip != 0 && isPrime(rp*rp + ip*ip)) )

    {

      char digits[maxDigits + 1];

      _itoa(n, digits, 4);

      cout<<digits << ", " << rp << (ip >= 0 ? "+" : "") << ip << "i\n";

    }

  }

}

CROSSREFS

A002145 when translated using A212494 is a subsequence.

Real and imaginary parts of the Gaussian primes A103431, A103432.

Sequence in context: A329761 A167612 A299974 * A116438 A191078 A281816

Adjacent sequences:  A262430 A262431 A262432 * A262434 A262435 A262436

KEYWORD

nonn,base

AUTHOR

Adam J.T. Partridge, Sep 22 2015

STATUS

approved

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Last modified July 2 08:50 EDT 2020. Contains 335398 sequences. (Running on oeis4.)