A262433 Quater-imaginary representation of the Gaussian primes with an even imaginary part.

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

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