A091100 -id:A091100 - cp's OEIS frontend

A295996 One quarter of number of Gaussian primes whose norm is 4*n+1 or less.

0, 3, 4, 6, 8, 8, 8, 10, 10, 12, 14, 14, 15, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 25, 27, 27, 29, 31, 31, 32, 32, 32, 32, 34, 34, 34, 36, 36, 38, 38, 38, 38, 40, 40, 42, 42, 42, 44, 46, 46, 46, 46, 46, 46, 46, 46, 48, 50, 50, 52, 52, 52, 52, 54, 54, 54, 56

Offset: 0

Seiichi Manyama, Dec 02 2017

nonn

			The Gaussian primes whose norm is 9 or less;
      *        3i,
    *   *      -1+2i, 1+2i
  * *   * *    -2+i, -1+i, 1+i, 2+i
*           *  -3, 3
  * *   * *    -2-i, -1-i, 1-i, 2-i
    *   *      -1-2i, 1-2i
      *        -3i
               a(2) = 16/4 = 4.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Gaussian Prime

Cf. A016813, A055029, A091100, A091134, A135462, A296020, A296021.

require 'prime'
def A(k, n)
  ary = []
  cnt = 0
  k.step(4 * n + k, 4){|i|
    cnt += 1 if i.prime?
    ary << cnt
  }
  ary
end
def A295996(n)
  ary1 = A(1, n)
  ary3 = A(3, Math.sqrt(n).to_i) + [0]
  [0] + (1..n).map{|i| 1 + 2 * ary1[i] + ary3[(Math.sqrt(4 * i + 1).to_i - 3) / 4]}
end
p A295996(100)

A091101 Number of first-quadrant Gaussian primes whose norm is less than 10^n.

Original entry on oeis.org

5, 27, 173, 1245, 9635, 78525, 664811, 5762247, 50850399, 455056167

Offset: 1

T. D. Noe, Dec 19 2003

Eric Weisstein's World of Mathematics, Gaussian Prime
Index entries for Gaussian integers and primes

Cf. A091098 (number of primes of the form 4k+1 less than 10^n), A091099 (number of primes of the form 4k+3 less than 10^n), A091100, A091102.

Table[lim2=10^n; lim1=Floor[Sqrt[lim2]]; cnt=0; Do[If[x^2+y^2True], cnt++ ], {x, 0, lim1}, {y, 0, lim1}]; cnt, {n, 6}]

a(2n) = 2*A091098(2n) + 2*A091099(n) + 1.

a(10) calculated from the data at A091098 and A091099 by Amiram Eldar, Feb 28 2020

A091134 Number of Gaussian primes a+b*i such that sqrt(a^2 + b^2) <= 10^n.

Original entry on oeis.org

100, 4928, 313752, 23046512, 1820205436, 150431552012, 12819767598972, 1116953361826164, 98959817242332844, 8883278410114778600

Offset: 1

Eric W. Weisstein, Dec 19 2003

Marc Deléglise, Pierre Dusart, and Xavier-Francois Roblot, Counting primes in residue classes, Math. Comp. 73 (2004), no. 247, 1565-1575
Eric Weisstein's World of Mathematics, Gaussian Prime

Bisection of A091100.

a(n) = 8*A091098(2*n) + 4*A091099(n) + 4. - Seiichi Manyama, Dec 03 2017

a(5)-a(10) from Seiichi Manyama using the data in A091098 and A091099, Dec 03 2017

cp's OEIS Frontend

A295996 One quarter of number of Gaussian primes whose norm is 4*n+1 or less.

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A091101 Number of first-quadrant Gaussian primes whose norm is less than 10^n.

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Mathematica

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A091134 Number of Gaussian primes a+b*i such that sqrt(a^2 + b^2) <= 10^n.

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