A296020 -id:A296020 - cp's OEIS frontend

A066490 Number of primes of the form 4m+3 that are <= n.

0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 1

Robert G. Wilson v, Jan 03 2002

Partial sums of A079261. - Reinhard Zumkeller, Feb 06 2014

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A000720, A066339, A296020.

a066490 n = a066490_list !! (n-1)
a066490_list = scanl1 (+) $ map a079261 [1..]
-- Reinhard Zumkeller, Feb 06 2014

Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 3 & ]], {n, 2, 100} ]
Accumulate[Table[If[PrimeQ[n]&&Mod[n,4]==3,1,0],{n,100}]] (* Harvey P. Dale, Mar 17 2021 *)

for(n=1,100,print1(sum(i=1,n,if((i*isprime(i)-3)%4,0,1)),","))

A066339(n) + a(n) = A000720(n) - 1 for n >= 2. - Jianing Song, Apr 28 2021

A296021 Number of primes of the form 4k+1 <= 4n+1.

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 26, 26, 27, 28, 28

Offset: 0

Seiichi Manyama, Dec 02 2017

nonn

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 23.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A016813, A091098, A295996, A296020.

a[n_]:=Length[Select[Range[n],PrimeQ[4#+1] &]]; Array[a,72,0] (* Stefano Spezia, May 01 2025 *)

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
p A(1, 100)

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

Original entry on oeis.org

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)

A340767 Number of primes p <= 6*n + 5 that are congruent to 2 modulo 3.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 13, 14, 15, 16, 16, 16, 17, 18, 18, 19, 19, 19, 20, 21, 22, 22, 23, 24, 24, 24, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 32, 33, 33, 33, 34, 35, 35, 35, 35, 35, 36, 37, 38, 38, 38, 38, 39, 40, 40, 41, 41, 41, 42, 42

Offset: 0

Jianing Song, Apr 28 2021

			There are 14 primes <= 6*16+5 = 101 that are congruent to 2 modulo 3, namely 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, so a(16) = 14.

Jianing Song, Table of n, a(n) for n = 0..10000

Cf. A003627, A167020, A340764, A172104, A296021, A296020.

a(n) = sum(i=1, 6*n+5, isprime(i) && (i%3==2))

a(n) = A340764(6*n+5).

a(n) = 1 + Sum_{k=0..n+1} A167020(k).

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A066490 Number of primes of the form 4m+3 that are <= n.

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A296021 Number of primes of the form 4*k+1 <= 4*n+1.

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

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Ruby

A340767 Number of primes p <= 6*n + 5 that are congruent to 2 modulo 3.

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Programs

PARI

Formula

A296021 Number of primes of the form 4k+1 <= 4n+1.