A106377 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime minus one.

1, 1, 2, 3, 2, 1, 4, 2, 1, 6, 7, 1, 10, 19, 25, 5, 13

Offset: 1

Author

Sven Simon, Apr 30 2005

Keywords

• nonn
• more

Comments

Consider the Gaussian primes of the first quadrant a+bi, with a > 0, b >= 0, ordered as a sequence by the size of the norm and the size of the real part a, as defined in A103431. The product of these primes up to a+bi, written here as cp#, may have the property that cp#+1 is a Gaussian prime. a(n) is the real part a of such a+bi. cp#+1 is not necessarily in the first quadrant.

From R. J. Mathar, Jun 13 2011: (Start)

Consider the partial products of the complex sequence A103431(n)+A103432(n)*i, which starts p# = 1+i, -1+3i, -5+5i, -15+15i, -75-15i, -195-195i, 585-975i, 3315-3315i, ... If 1+p# is a Gaussian prime, we insert the real part of the last factor, A103431(n), into this sequence. The first missing element is A103431(6), meaning -194-195i is not a Gaussian prime.

The 7 is for products up to norm 192, the 1 for products up to 256, the 10 for 268, 19 up to 360 and the 25 up to 820. (No further up to norm 5700. Is the sequence finite?) (End)

Examples

			(1+i)*(1+2i)*(2+i)*3*(2+3i) + 1 = (-75-15i) + 1 = (-74-15i), which is a Gaussian prime. This is the 5th number with the property, so a(5) = 2.
		

Links

• Sven Simon, List for A106377/A106378.

Crossrefs

Cf. A103431, A103432, A106378, A106379, A106381, A106383.

Extensions

Offset corrected and a(16)-a(17) added by Amiram Eldar, Aug 16 2025