A103432 -id:A103432 - cp's OEIS frontend

A103431 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. a(n) is the real part of the Gaussian prime. Sequence A103432 gives the imaginary parts.

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

Offset: 1

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

nonn

Definition of Gaussian primes (Pieper, Die komplexen Zahlen, p. 122): 1) i+i, norm N(i+i) = 2. 2) Natural primes p with p = 3 mod 4, norm N(p) = p^2. 3) primes a+bi, a>0, b>0 with a^2 + b^2 = p = 1 mod 4, p natural prime. Norm N(a+bi) = p. b+ai is a different Gaussian prime number, b+ai cannot be factored into a+bi and a unit. 4) All complex numbers from 1) to 3) multiplied by the units -1,i,-i, these are the associated numbers. The sequence contains all the Gaussian primes mentioned in 1) - 3).
Every complex number can be factored completely into the Gaussian prime numbers defined by the sequence, an additional unit as factor can be necessary. This factorization can be used to calculate the complex sigma, as defined by Spira. The elements a(n) are ordered by the size of their norm. If the two different primes a+bi and b+ai have the same norm, they are ordered by the size of the real part of the complex prime number. So a+bi follows b+ai in the sequence, if a > b.
Of course this is not the only possible definition. As primes p = 1 mod 4 can be factored in p = (-i)(a+bi)(b+ai) and the norm N(a+bi) = N(b+ai) = p, these primes a+bi occur much earlier in the sequence than p does in the sequence of natural primes. 4+5i with norm 41 occurs before prime 7 with norm 49.

H. Pieper, "Die komplexen Zahlen", Verlag Harri Deutsch, p. 122

Robert Israel, Table of n, a(n) for n = 1..10000
Sven Simon, List with Gaussian primes (extended) of A103431/A103432
R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.
Wikipedia, Table of Gaussian integer factorizations

Cf. A103432, A055025.

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):
g:= proc(z)
    local a,b;
    a:= Re(z); b:= Im(z);
    if b = 0 then z
    else
      a:= abs(a);
      b:= abs(b);
      if a = b then a
      elif a < b then a,b
      else b,a
      fi
    fi
end proc:
map(g, 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[#]]&] // Re (* Jean-François Alcover, Mar 04 2019 *)

Edited (mostly to correct meaning of norm) by Franklin T. Adams-Watters, Mar 04 2011

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

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

A106381 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime minus i.

Original entry on oeis.org

1, 1, 2, 2, 1, 6, 4, 11, 10, 11, 19, 3, 18, 16, 40, 27, 139

Offset: 1

Sven Simon, Apr 30 2005

A106382 has the imaginary parts of these numbers.

			(1+i)*(1+2i)*(2+i)*3*(2+3i)*(3+2i)*(1+4i) + i = (585-975i) + i = (585-974i), which is a Gaussian prime. This is the 5th number with the property, so a(5) = 1.

Sven Simon, Readable list of A106381/A106382.

Cf. A103431, A103432, A106377, A106379, A106382, A106384.

a(15)-a(17) from Amiram Eldar, Aug 16 2025

A106383 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime plus i.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 2, 6, 5, 5, 5, 4, 1, 25, 20, 3, 29, 36, 74, 112, 140, 48

Offset: 1

Sven Simon, Apr 30 2005

A106384 has the imaginary parts.

			(1+i)*(1+2i)*(2+i)*3*(2+3i)*(3+2i)*(1+4i)*(4+i)*(2+5i) - i = (23205+9945i) - i = (23205+9944i), which is a Gaussian prime. This is the 7th number with the property, so a(7) = 2.

Sven Simon, Readable list of A106383/A106384.

Cf. A103431, A103432, A106377, A106379, A106381, A106384.

a(18)-a(22) from Amiram Eldar, Aug 16 2025

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.

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

Original entry on oeis.org

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

Adam J.T. Partridge, Sep 22 2015

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.

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

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

A002145 when translated using A212494 is a subsequence.

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

cp's OEIS Frontend

A103431 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. a(n) is the real part of the Gaussian prime. Sequence A103432 gives the imaginary parts.

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

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

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A106381 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime minus i.

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A106383 Real part of Gaussian prime numbers such that the Gaussian primorial product up to them is a Gaussian prime plus i.

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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.

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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.

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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.

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

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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.