A296021 -id:A296021 - cp's OEIS frontend

A066339 Number of primes p of the form 4m+1 with p <= n.

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

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 01 2002

nonn

Asymptotic expansion: a(n) ~ pi(n)/2 ~ n/(2log(n)) (pi(n) is in sequence A000720).

Partial sums of A079260. - Reinhard Zumkeller, Feb 06 2014

T. D. Noe, Table of n, a(n) for n=1..10000
R. Breusch, An Asymptotic Formula for Primes Of The Form 4n+1

Cf. A000720, A066490, A296021.

a066339 n = a066339_list !! (n-1)
a066339_list = scanl1 (+) $ map a079260 [1..]
-- Reinhard Zumkeller, Feb 06 2014

Table[ Length[ Select[ Union[ Table[ Prime[ PrimePi[i]], {i, 2, n}]], Mod[ #, 4] == 1 & ]], {n, 2, 100} ]

for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-1)%4,0,1)),","))

a(n) + A066490(n) = A000720(n) - 1 for n >= 2. - Jianing Song, Apr 28 2021

More terms from Robert G. Wilson v, Jan 03 2002

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)

A296020 Number of primes of the form 4k+3 <= 4n+3.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 10, 11, 11, 12, 13, 13, 13, 13, 13, 14, 15, 15, 15, 15, 15, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 25, 26, 27, 27, 27, 28, 28, 28, 29, 29, 29, 30, 30, 31, 31, 31

Offset: 0

Seiichi Manyama, Dec 02 2017

nonn

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

Cf. A004767, A091099, A295996, A296021.

Module[{nn=300,pr3},pr3=Table[If[PrimeQ[k]&&Mod[k,4]==3,1,0],{k,0,nn}];Table[Total[Take[pr3,4n+3]],{n,(nn-3)/4}]] (* Harvey P. Dale, Aug 10 2019 *)

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(3, 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).

A307152 a(n) = floor((A002144(n)+19)/24).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 31 2019

nonn

This sequence arises in several different contexts [Kramer].
The number of occurrences of k in the sequence is A296021(6*k) - A296021(6*k-6). - Robert Israel, Mar 31 2019
Original name was: "Floor( (q+19)/24 ) where q is a prime == 1 (mod 4)." - Robert Israel, Apr 07 2019

Robert Israel, Table of n, a(n) for n = 1..10000
Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Cf. A002144, A296021.

map(t -> floor((t+19)/24), select(isprime, [seq(i,i=1..1000,4)])); # Robert Israel, Mar 31 2019

Table[Floor[(q + 19)/24], {q, Select[Range[1, 650, 4], PrimeQ]}] (* Michael De Vlieger, Mar 31 2019 *)

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A066339 Number of primes p of the form 4m+1 with p <= n.

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

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A296020 Number of primes of the form 4*k+3 <= 4*n+3.

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

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A307152 a(n) = floor((A002144(n)+19)/24).

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A296020 Number of primes of the form 4k+3 <= 4n+3.