A103431 -id:A103431 - cp's OEIS frontend

A103432 Subsequence of the Gaussian primes, where only Gaussian primes a+bi with a>0, b>=0 are listed. Ordered by the norm N(a+bi)=a^2+b^2 and the size of the real part when the norms are equal. The sequence gives the imaginary parts. See A103431 for the real parts.

1, 2, 1, 0, 3, 2, 4, 1, 5, 2, 6, 1, 5, 4, 0, 7, 2, 6, 5, 8, 3, 8, 5, 9, 4, 10, 1, 10, 3, 8, 7, 0, 11, 4, 10, 7, 11, 6, 13, 2, 10, 9, 12, 7, 14, 1, 15, 2, 13, 8, 15, 4, 16, 1, 13, 10, 14, 9, 16, 5, 17, 2, 13, 12, 14, 11, 16, 9, 18, 5, 17, 8, 0, 18, 7, 17, 10, 19, 6, 20, 1, 20, 3, 15, 14, 17

Offset: 1

Sven Simon, Feb 05 2005; corrected Feb 20 2005 and again on Aug 06 2006

nonn

Detailed description in A103431.

Robert Israel, Table of n, a(n) for n = 1..10000
Sven Simon, List with Gaussian primes of A103431/A103432

N:= 100: # to get all terms with norm <= N
p1:= select(isprime,[seq(i,i=3..N,4)]):
p2:= select(isprime,[seq(i,i=1..N^2,4)]):
p2:= map(t -> GaussInt:-GIfactors(t)[2][1][1],p2):
p3:= sort( [1+I, op(p1),op(p2)],(a,b) -> Re(a)^2 + Im(a)^2  < Re(b)^2 + Im(b)^2):
h:= proc(z)
    local a,b;
    a:= Re(z); b:= Im(z);
    if b = 0 then 0
    else
      a:= abs(a);
      b:= abs(b);
      if a = b then a
      elif a < b then b,a
      else a,b
      fi
    fi
end proc:
map(h,p3); # Robert Israel, Feb 23 2016

maxNorm = 500;
norm[z_] := Re[z]^2 + Im[z]^2;
m = Sqrt[maxNorm] // Ceiling;
gp = Select[Table[a + b I, {a, 1, m}, {b, 0, m}] // Flatten, norm[#] <= maxNorm && PrimeQ[#, GaussianIntegers -> True]&];
SortBy[gp, norm[#] maxNorm + Abs[Re[#]]&] // Im (* Jean-François Alcover, Feb 26 2019 *)

Definition of norm corrected by Franklin T. Adams-Watters, Mar 04 2011

a(48) corrected by Robert Israel, Feb 23 2016

A107632 Subsequence of A107629. Consider a Gaussian prime a+bi with index k in A103431. k is in A107632 when an integer multiplier m exists such that the distance of mnorm(a+bi) to k is minimal up to k. abs(mnorm(a+bi) - k) is minimal up to k. A107633 gives the squares of the norms of these Gaussian primes, A107634 the integer multipliers m.

Original entry on oeis.org

1, 2, 12, 80, 218, 447, 448, 590, 955, 4657, 6787, 63041, 127337, 3886223, 11862335, 41822073

Offset: 1

Sven Simon, May 18 2005

			The Gaussian prime 19411+20906i has index 41822073 in A103431. Norm(19411+20906i) = 28528.01705341..., square of norm is 813847757 and multiplier m = 1466. sqrt(813847757)*1466 = 41822073.00028..., a(16)=41822073.

Cf. A103431, A103432, A107629, A107633, A107634.

A107629 The present sequence depends on the index k of a Gaussian prime a + bi in A103431. Such an index k is a term of this sequence when an integer multiplier m exists such that mnorm(a+bi) lies in an interval of length 1 around the index k of a+bi in A103431: k - 1/2 < mnorm(a+bi) < k + 1/2.

Original entry on oeis.org

1, 2, 8, 12, 13, 38, 39, 80, 142, 143, 216, 218, 221, 222, 325, 329, 330, 447, 448, 450, 590, 594, 765, 954, 955, 1156, 1413, 1418, 1419, 1658, 1660, 1661, 1666, 1667, 1958, 2259, 2260, 2590, 2595, 2940, 3340, 3342, 3763, 4209, 4656, 4657, 4662, 4663, 4668

Offset: 1

Sven Simon, May 18 2005

nonn

Consider the Gaussian primes a + bi of the first quadrant ordered as a sequence as in A103431. In A103431 and A103432 these primes are ordered first by their norm and if the norms are equal, by the size of the real part a. A prime p == 1 (mod 4) splits into two different Gaussian primes p = -i(a+bi)(b+ai) where a^2 + b^2 = p and these two primes have the same norm. Through this kind of ordering the primes have a well-defined index k in A103431. The present sequence depends on the index k of a Gaussian prime a + bi in A103431. Such an index k is a term of this sequence when an integer multiplier m exists such that m*norm(a+bi) lies in an interval of length 1 around the index k of a+bi in A103431: k - 1/2 < m*norm(a+bi) < k + 1/2. Counting roughly the first 50000000 Gaussian primes of A103431, every integer < 1600 appeared at least once as a multiplier.

As this property depends only on the norm, one could choose for example the Gaussian primes of the 4th quadrant and would get the same results. It is only necessary that no Gaussian primes are included which are multiples of each other and a unit (-1,i,-i). A107630 gives the squares of the norms, which are integers. A107631 gives the multipliers m. Sequence A107632 (cf. also A107633, A107634) is a subsequence of the present sequence where the distance m*norm(a+bi) from index k is smaller than for all previous values, abs(m*norm(a+bi)-k) is minimal up to k.

			The Gaussian prime with index k=80 in sequence A103431 is 1+20i, norm(1+20i)=20.0249..., norm(1+20i)^2=401. With multiplier m = 4, 4*norm(1+20i) = 80.0999375..., which is in the interval with length 1 around 80. So a(8)=80.

Cf. A103431, A103432, A107630, A107631, A107632, A107633, A107634.

A106385 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 a, as defined in A103431. The sum of these primes up to a+bi is divisible by a+bi. a(n) is the real part a of such a+bi.

Original entry on oeis.org

1, 2, 3, 1, 191

Offset: 1

Sven Simon, Apr 30 2005

nonn

A106386 has the imaginary parts.

			(1+i)+(1+2i)+(2+i)+3+(2+3i)+(3+2i)+(1+4i)+(4+i)+(2+5i)+(5+2i)+(1+6i)+(6+i)+(4+5i)+
(5+4i)+7+(2+7i)+(7+2i)+(5+6i)+(6+5i)+(3+8i) = (70+65i), (70+65i) / (3+8i) = (10-5i).
This is the third number with the property, so a(3)=3.

Sven Simon, Table of n, a(n), A106385(n), a+b*i and the associated partial sum of A103431

Cf. A103431, A103432, A106386.

A122435 Complex numbers a+bi which are divisible by the sum of their complex prime factors with repetition (complex sopfr). The prime factors are from the first quadrant only, as defined in A103431. This sequence gives the real part a of these numbers, the imaginary part b is in A122436.

Original entry on oeis.org

10, 15, 16, 27, 30, 17, 34, 39, 14, 38, 14, 44, 48, 39, 40, 1, 57, 45, 54, 28, 64, 21, 46, 57, 72, 45, 75, 44, 64, 38, 70, 4, 84, 72, 60, 70, 78, 100, 53, 89, 52, 90, 5, 105, 6, 42, 104, 70, 90, 114

Offset: 1

Sven Simon, Sep 04 2006

nonn

In the case of the complex sopfr it seems best to use only primes in the first quadrant because it is easy to get a well-defined function.

			a(1) = 10 is the real part of 10 = (-1)*(1+i)*(1+i)*(1+2i)*(2+i) = (1-i)(5+5i); (1+i)+(1+i)+(1+2i)+(2+i) = 5+5i;
a(5) = 30 is the real part of 30+12i = (-i)*3*(1+i)*(1+i)*(5+2i) = 3*(10+4i); 3+(1+i)+(1+i)+(5+2i) = 10+4i.

Sven Simon, Readable list of A122435/A122436
R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.

Cf. A103431, A103432.

A190635 Index of A190637(n) in the sequence A103431 of Gaussian primes.

Original entry on oeis.org

4, 280, 3469989, 34317382, 73414350, 178935505

Offset: 1

Sven Simon, May 15 2011

nonn

			A190637(2)=43 is the a(2)=280th entry in the sequence A103431 (which orders the Gaussian primes of the first quadrant).

Cf. A103431, A190637, A190634.

a(5)-a(6) from Sven Simon, Jun 19 2011

A190637 Primes p == 3 mod 4 whose index as prime divides their index as a Gaussian prime (in the first quadrant, as defined in A103431, for example).

Original entry on oeis.org

3, 43, 7639, 25703, 38371, 61291

Offset: 1

Sven Simon, May 15 2011

The index of a prime p = 3 mod 4 as a Gaussian prime is well defined, it is summed up by 1 for the complex prime 1+i (as factor of prime 2 = -i*(1+i)^2).
The count of primes (3 mod 4) <= p, which remain unchanged as they cannot be factored further into complex primes 2 times the count of primes (1 mod 4) <= p**2 (such primes p1 are split into two distinct complex primes of the first quadrant with size sqrt(p1)).
As the result from the splitting of the primes 1 mod 4, the indices of primes 3 mod 4 as Gaussian prime grows rather rapidly against their index as normal prime.
Interesting numerical effects: the prime index of 43 is 14, with 3*14+1 = 43. 43 is the upper part of twin prime with 41 (which would be 14*3 - 1 with an index 14, if 1 was counted as prime). 4241 and 4243 are both primes.
The ratio f between both indices can be estimated as f = (p^2 / log(p^2)) / (p / log(p)) = p/2. - Sven Simon, May 26 2011

			The prime 3 has index 2, as a Gaussian prime it has index 4 (the list is 1+i, 1+2i, 2i+1, 3, ...), and 2 divides 4.

Cf. A103431 (Gaussian primes in first quadrant), A190634 (prime index), A190635 (index as Gaussian prime).

Changed name definition which was a bit wrong, the index is not a prime number

A345436 Represent the ring of Gaussian integers E = {x+y*i: x, y rational integers, i = sqrt(-1)} by the cells of a square grid; number the cells of the grid along a counterclockwise square spiral, with the cells representing the ring identities 0, 1 numbered 0, 1. Sequence lists the index numbers of the cells which are 0 or a prime in E.

Original entry on oeis.org

0, 2, 4, 6, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 59, 61, 67, 69, 75, 77, 81, 83, 87, 89, 91, 93, 97, 99, 101, 103, 107, 109, 111, 113, 117, 119, 121, 125, 127, 131, 133, 137, 139, 143, 145, 149, 151, 155, 157

Offset: 1

N. J. A. Sloane, Jun 23 2021

nonn

The cell with spiral index m represents the Gaussian integer A174344(m+1) + A274923(m+1) * i. So the set of Gaussian primes is {A174344(a(n)+1) + A274923(a(n)+1) * i : n >= 2}. - Peter Munn, Aug 02 2021
The Gaussian integer z = x+i*y has norm x^2+y^2. There are four units (of norm 1), +-1, +-i. The number of Gaussian integers of norm n is A004018(n).
The norms of the Gaussian primes are listed in A055025, and the number of primes with a given norm is given in A055026.
The successive norms of the Gaussian integers along the square spiral are listed in A336336.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag; Table 4.2, p. 106.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173.

Eric Weisstein's World of Mathematics, Gaussian prime.
Brian Wichmann, Tiling for Unique Factorization Domains, Jul 22 2019. See Fig. 2.

Equals A308412 - 1. Cf. A345435, A345437.

Cf. also A003173, A004018, A055025, A055026, A103431, A174344, A274923, A336336.

Name clarified by Peter Munn, Aug 02 2021

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

Original entry on oeis.org

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

Offset: 1

Sven Simon, Apr 30 2005

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)

			(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.

Sven Simon, List for A106377/A106378.

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

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

A106379 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime plus one.

Original entry on oeis.org

1, 2, 3, 6, 5, 11, 10, 18, 12, 19, 10, 13, 5, 20, 6, 50, 74, 112, 40, 140, 139

Offset: 1

Sven Simon, Apr 30 2005

A106380 has the imaginary parts.

			(1+i)*(1+2i)*(2+i)*3*(2+3i)*(3+2i) - 1 = (-195-195i) - 1 = (-196-195i), which is a Gaussian prime. This is the third number with the property, so a(3) = 3.

Sven Simon, Readable list for A106379/A106380.

Cf. A103431, A103432, A106377, A106380, A106381, A106383.

Offset corrected and a(16)-a(21) added by Amiram Eldar, Aug 16 2025

cp's OEIS Frontend

A103432 Subsequence of the Gaussian primes, where only Gaussian primes a+bi with a>0, b>=0 are listed. Ordered by the norm N(a+bi)=a^2+b^2 and the size of the real part when the norms are equal. The sequence gives the imaginary parts. See A103431 for the real parts.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Extensions

Views

Author

Keywords

Examples

Crossrefs

A107629 The present sequence depends on the index k of a Gaussian prime a + bi in A103431. Such an index k is a term of this sequence when an integer multiplier m exists such that m*norm(a+bi) lies in an interval of length 1 around the index k of a+bi in A103431: k - 1/2 < m*norm(a+bi) < k + 1/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

A106385 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 a, as defined in A103431. The sum of these primes up to a+bi is divisible by a+bi. a(n) is the real part a of such a+bi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A122435 Complex numbers a+bi which are divisible by the sum of their complex prime factors with repetition (complex sopfr). The prime factors are from the first quadrant only, as defined in A103431. This sequence gives the real part a of these numbers, the imaginary part b is in A122436.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A190635 Index of A190637(n) in the sequence A103431 of Gaussian primes.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A190637 Primes p == 3 mod 4 whose index as prime divides their index as a Gaussian prime (in the first quadrant, as defined in A103431, for example).

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A106379 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime plus one.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A107629 The present sequence depends on the index k of a Gaussian prime a + bi in A103431. Such an index k is a term of this sequence when an integer multiplier m exists such that mnorm(a+bi) lies in an interval of length 1 around the index k of a+bi in A103431: k - 1/2 < mnorm(a+bi) < k + 1/2.