A055025 -id:A055025 - cp's OEIS frontend

A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

10, 2, 2, 5, 5, 130, 185, 5, 2, 2, 10, 10, 5, 5, 10, 17, 17, 5, 5, 5, 53, 53, 13, 13, 1490, 5, 2, 2, 5, 1565, 5

Offset: 1

Artur Jasinski, Nov 29 2011

Conjecture (Jasiński): The numbers in this sequence are multiplicative combinations of: primes congruent to 1 or 2 modulo 4 (A002313), Pythagorean primes (A002144), the number 2, and norms of Gaussian primes A055025.

Cf. A200216, A200656, A201047.

Minor edits by N. J. A. Sloane, Feb 23 2014

A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

Two primes are considered equivalent if they differ by multiplication by a unit (+-1, +-omega, +-omega^2).

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

norms = Join[{3}, Select[Range[2000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Length[Reduce[n == a^2 - a*b + b^2, {a, b}, Integers]]/6; A055666 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

More terms from Franklin T. Adams-Watters, May 05 2006

A218858 Number of Gaussian primes at taxicab distance n from the origin.

Original entry on oeis.org

0, 0, 4, 12, 0, 16, 0, 20, 0, 16, 0, 28, 0, 24, 0, 32, 0, 32, 0, 36, 0, 24, 0, 36, 0, 64, 0, 32, 0, 48, 0, 44, 0, 32, 0, 72, 0, 64, 0, 48, 0, 72, 0, 60, 0, 56, 0, 60, 0, 40, 0, 56, 0, 72, 0, 112, 0, 64, 0, 76, 0, 88, 0, 56, 0, 136, 0, 92, 0, 80, 0, 76, 0, 88, 0

Offset: 0

T. D. Noe, Nov 12 2012

nonn

Except for n = 2, there are no Gaussian primes at an even taxicab distance from the origin. All terms are multiples of 4. See A218859 for this sequence divided by 4.

The arithmetic derivative of Gaussian primes is either 1, -1, I, or -I.

			In the taxicab distance, the four Gaussian primes closest to the origin are 1+I, -1+I, -i-I, and 1-I. The 12 at taxicab distance 3 are the four reflections of 3, 2+I, and 1+2I.

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Linear plot

Cf. A055025 (norms of Gaussian primes).
Cf. A222593 (first-quadrant Gaussian primes).
Cf. A225071, A225072 (number of terms at an odd distance from the origin).

Table[cnt = 0; Do[If[PrimeQ[n - i + I*i, GaussianIntegers -> True], cnt++], {i, 0, n}]; Do[If[PrimeQ[i - n + I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 0, -1}]; Do[If[PrimeQ[i - n - I*i, GaussianIntegers -> True], cnt++], {i, 1, n}]; Do[If[PrimeQ[n - i - I*i, GaussianIntegers -> True], cnt++], {i, n - 1, 1, -1}]; cnt, {n, 0, 100}]

A239397 Prime Gaussian integers x + y*i sorted by norm and increasing y, with x and y nonnegative.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 0, 0, 3, 3, 2, 2, 3, 4, 1, 1, 4, 5, 2, 2, 5, 6, 1, 1, 6, 5, 4, 4, 5, 7, 0, 0, 7, 7, 2, 2, 7, 6, 5, 5, 6, 8, 3, 3, 8, 8, 5, 5, 8, 9, 4, 4, 9, 10, 1, 1, 10, 10, 3, 3, 10, 8, 7, 7, 8, 11, 0, 0, 11, 11, 4, 4, 11, 10, 7, 7, 10, 11, 6, 6, 11, 13, 2

Offset: 1

T. D. Noe, Mar 22 2014

After the number 1 + i, there are exactly two Gaussian primes here for each norm in A055025; if x + y*i is here, then y + x*i is also. - T. D. Noe, Mar 26 2014

Sequence A239621 provides a more condensed version, without y + x*i following each x + y*i. The real parts and imaginary parts are listed in A300587 and A300588. - M. F. Hasler, Mar 09 2018

			The sequence of Gaussian primes (with nonnegative real and imaginary part) begins 1+i, 2+i, 1+2i, 3, 3i,...

T. D. Noe, Table of n, a(n) for n = 1..10002 (5001 complex numbers)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Complex Number

Cf. A055025 (norms of Gaussian primes), A239621, A300587, A300588.

mx = 20; lst = Flatten[Table[{a, b}, {a, 0, mx}, {b, 0, mx}], 1]; qq = Select[lst, Norm[#] <= mx && PrimeQ[#[[1]] + I*#[[2]], GaussianIntegers -> True] &]; Sort[qq, Norm[#1] < Norm[#2] &]

is_GP(x,y=0)={(x=factor(if(imag(x+I*y),x+I*y,I*x+y)))&&vecsum(x[,2])==1+(abs(x[1,1])==1)} \\ Returns 1 iff x + iy (y may be omitted) is a Gaussian prime. -  M. F. Hasler, Mar 10 2018

for(N=2,499, if(isprime(N) && N%4<3, z=factor(I*N); for(i=0,N>2, print1(real(z[i+1,1])","imag(z[i+1,1])",")), issquare(N,&z) && isprime(z) && z%4==3 && print1(z",0,0,"z","))) \\ M. F. Hasler, Mar 10 2018

a(4n + 1) = a(4n) = A239621(2n) = A300588(n), a(4n + 2) = a(4n-1) = A239621(2n-1) = A300587(n). - M. F. Hasler, Mar 09 2018

A239621 Gaussian primes x + i*y, with x = a(2n-1) >= y = a(2n) >= 0, sorted by norm.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 3, 2, 4, 1, 5, 2, 6, 1, 5, 4, 7, 0, 7, 2, 6, 5, 8, 3, 8, 5, 9, 4, 10, 1, 10, 3, 8, 7, 11, 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, 19, 0

Offset: 1

T. D. Noe, Mar 22 2014

nonn

The condition a >= b >= 0 implies that there is only one Gaussian prime for each norm. - T. D. Noe, Mar 26 2014

The real parts and imaginary parts are listed as a(2n-1) = A300587(n) and a(2n) = A300588(n), respectively. Sequence A239397 lists the pair (y, x) after each pair (x, y), except for (1, 1). - M. F. Hasler, Mar 10 2018

			From _M. F. Hasler_, Mar 09 2018: (Start)
Sorted by norm, the smallest Gaussian primes z = x + iy in the first half-quadrant x >= y >= 0 are:
a(1) + i*a(2) = 1 + i;
a(3) + i*a(4) = 2 + i;
a(5) + i*a(6) = 3;
... (End)

T. D. Noe, Table of n, a(n) for n = 1..10106 (5053 pairs)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Complex Number

Cf. A055025 (norms of Gaussian primes), A239397.

Cf. A300587, A300588.

mx = 20; lst = Flatten[Table[{a, b}, {a, 0, mx}, {b, 0, a}], 1]; qq = Select[lst, Norm[#] <= mx && PrimeQ[#[[1]] + I*#[[2]], GaussianIntegers -> True] &]; Sort[qq, Norm[#1] < Norm[#2] &]

{for(n=2,400, f=factor(n*I)/*factor in Z[i]*/; matsize(f)[1]<=2 && vecsum(f[,2])==2+(f[1,1]==I) /*either I*p^2 or w*conj(w/I), maybe (1+I)^2 */ && printf("%d,",vecsort([real(f=f[3-f[1,2],1]),imag(f)],,4)))} \\ For illustrative use. - M. F. Hasler, Mar 09 2018

Name changed and in cf. complex -> Gaussian - Wolfdieter Lang, Mar 25 2014

Name edited by M. F. Hasler, Mar 09 2018

A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Original entry on oeis.org

6, 6, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 6, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12

Offset: 1

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+b*omega, a and b rational integers, omega = (1+sqrt(-3))/2.

			There are 6 Eisenstein-Jacobi primes of norm 3, omega-omega^2 times one of the 6 units [ +-1, +-omega, +-omega^2 ] but only one up to equivalence.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. VI.

Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-file of A055666 with the formula of _Franklin T. Adams-Watters_)

Cf. A055664-A055668, A055025-A055029. See A004016 and A035019 for theta series of Eisenstein (or hexagonal) lattice.

norms = Join[{3}, Select[Range[1000], (PrimeQ[#] && Mod[#, 6] == 1) || (PrimeQ[Sqrt[#]] && Mod[Sqrt[#], 3] == 2) &]]; r[n_] := Reduce[n == a^2 - a*b + b^2, {a, b}, Integers] // Length; A055665 = r /@ norms (* Jean-François Alcover, Oct 24 2013 *)

a(n) = 6 * A055666(n) - Franklin T. Adams-Watters, May 05 2006

More terms from Franklin T. Adams-Watters, May 05 2006

A300587 Real part of the n-th Gaussian prime x + i*y, x >= y >= 0, ordered by norm x^2 + y^2.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 6, 5, 7, 7, 6, 8, 8, 9, 10, 10, 8, 11, 11, 10, 11, 13, 10, 12, 14, 15, 13, 15, 16, 13, 14, 16, 17, 13, 14, 16, 18, 17, 19, 18, 17, 19, 20, 20, 15, 17, 20, 21, 19, 22, 20, 23, 21, 19, 20, 24, 23, 24, 18, 19, 25, 22, 25, 23, 26, 26, 22, 27, 26, 20

Offset: 1

M. F. Hasler, Mar 09 2018

With the restriction Re(z) >= Im(z) >= 0 used here and in A239621, there is exactly one Gaussian prime z for each possible norm |z|^2 in A055025. Sequence A239397 lists both, (x, y) and (y, x), for each of these having x > y (i.e., except for x = y = 1).

The nice graph shows that the values are denser towards the upper bound a(n) <= sqrt(A055025(n)) ~ sqrt(2n log n) than to the lower bound sqrt(A055025(n)/2) ~ sqrt(n log n), while for the imaginary parts A300588, i.e., min(Re(z),Im(z)), the distribution looks rather uniform.

M. F. Hasler, Table of n, a(n) for n = 1..10000

Odd bisection of A239621. See A300588 for imaginary parts, A055025 for the norms.

c=1; for(n=1,oo, matsize(f=factor(n*I))[1]<=2 && vecsum(f[,2])==2+(f[1, 1]==I) && !write("/tmp/b300587.txt",c" "max(real(f=f[3-f[1,2],1]),imag(f))) && c++>1e4 && break) \\ Replace write("/tmp/b300587.txt",c" by print1(", to print the values.

a(n) = A239621(2n-1) = A239397(4n-2) (= A239397(4n-5) for n > 1).

a(n) = sqrt(A055025(n) - A300588(n)^2).

A055028 Number of Gaussian primes of norm n.

Original entry on oeis.org

0, 0, 4, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0

Offset: 0

N. J. A. Sloane, Jun 09 2000

These are the primes in the ring of integers a+bi, a and b rational integers, i = sqrt(-1).

			There are 8 Gaussian primes of norm 5, +-1 +- 2i and +-2 +- i, but only two inequivalent ones (2 +- i).

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for Gaussian integers and primes

Cf. A055025-A055029, A055664-... , A295996.

A055028 := proc(n::integer)
    local c,a,b ;
    c := 0 ;
    for a from -n to n do
        if issqr(n-a^2) then
            b := sqrt(n-a^2) ;
            if GaussInt[GIprime](a+b*I) and a^2+b^2=n then
                if b = 0 then
                    c := c+1 ; # a+i*b and a-i*b
                else
                    c := c+2 ; # a+i*b and a-i*b
                end if;
            end if;
        end if;
    end do:
    c ;
end proc:
seq( A055028(n),n=0..50) ; # R. J. Mathar, Jul 22 2021

a[n_ /; PrimeQ[n] && Mod[n, 4] == 1] = 8; a[2] = 4; a[n_ /; (p = Sqrt[n]; PrimeQ[p] && Mod[p, 4] == 3)] = 4; a[] = 0; Table[ a[n], {n, 0, 100}] (* _Jean-François Alcover, Jul 30 2013, after Franklin T. Adams-Watters *)

a(n) = 4 * A055029(n). - Franklin T. Adams-Watters, May 05 2006

More terms from Reiner Martin, Jul 20 2001

A319920 Let f(1) = 1 + i (where i denotes the imaginary unit) and for n > 0, f(n+1) is the Gaussian prime in the first quadrant (with positive real part and nonnegative imaginary part) with least modulus that divides 1 + Product_{k=1..n} f(k) (in case of a tie minimize the imaginary part); a(n) is the square of the modulus of f(n).

Original entry on oeis.org

2, 5, 13, 9, 1129, 29, 17, 651250309, 5, 13, 17, 29, 37, 16767128453, 41, 133981, 2236369, 61, 45293, 22481146745713207066897, 12041, 653, 51908348513173, 121, 11821, 779353

Offset: 1

Rémy Sigrist, Oct 01 2018

The real and imaginary parts of f are respectively given by A320103 and A320104.
This sequence is a complex variant of the Euclid-Mullin sequence (A000945).
All terms belong to A055025.

Rémy Sigrist, PARI program for A319920

Cf. A000945, A055025, A320103, A320104.

PARI
```
See Links section.
```

a(n) = A320103(n)^2 + A320104(n)^2.

A344123 Decimal expansion of Sum_{i > 0} 1/A001481(i)^2.

Original entry on oeis.org

1, 4, 2, 6, 5, 5, 6, 0, 6, 3, 5, 1, 2, 5, 9, 2, 8, 7, 8, 6, 9, 6, 8, 0, 9, 3, 1, 6, 1, 5, 5, 0, 8, 1, 6, 3, 6, 1, 2, 7, 6, 6, 9, 3, 6, 3, 6, 7, 7, 0, 3, 9, 0, 2, 8, 8, 7, 9, 9, 2, 2, 3, 0, 4, 4, 1, 2, 9, 6, 0, 4, 5, 2, 8, 6, 1, 5, 1, 9, 0, 1, 9, 1, 4, 6, 7

Offset: 1

A.H.M. Smeets, May 09 2021

This constant can be considered as an equivalent of zeta(2) (= Pi^2/6 = A013661), where Euler's zeta(2) is over all positive integers, with prime elements in A000040, while this constant is over all positive integers that can be written as the sum of two squares (A001481) with prime elements given in A055025.

Close to the value of e^(3/2)/Pi.

			1.4265560635125928786968093161550816361276693636770...

R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.

Cf. A000040, A001481, A055025, A175647, A243379, A334448.

Cf. A344124, A344125.

Equals Sum_{i > 0} 1/A001481(i)^2.
Equals Product_{i > 0} 1/(1-A055025(i)^-2).
Equals 1/(1-prime(1)^(-2)) * Product_{i>1 and prime(i) == 1 (mod 4)} 1/(1-prime(i)^(-2)) * Product_{i>1 and prime(i) == 3 (mod 4)} 1/(1-prime(i)^(-4)), where prime(n) = A000040(n).
Equals (4/3)/(A243379*A334448).
Equals zeta_{2,0} (2) * zeta_{4,1} (2) * zeta_{4,3} (4), where zeta_{4,1} (2)  = A175647 and zeta_{2,0} (s) = 2^s/(2^s - 1).

cp's OEIS Frontend

A201278 a(n) specifies the quadratic extension sqrt(a(n)) for A201047(n).

Views

Author

Keywords

Comments

Crossrefs

Extensions

A055666 Number of inequivalent Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

A218858 Number of Gaussian primes at taxicab distance n from the origin.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A239397 Prime Gaussian integers x + y*i sorted by norm and increasing y, with x and y nonnegative.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A239621 Gaussian primes x + i*y, with x = a(2n-1) >= y = a(2n) >= 0, sorted by norm.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A055665 Number of Eisenstein-Jacobi primes of successive norms (indexed by A055664).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A300587 Real part of the n-th Gaussian prime x + i*y, x >= y >= 0, ordered by norm x^2 + y^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A055028 Number of Gaussian primes of norm n.

Views