A091099 -id:A091099 - cp's OEIS frontend

A091098 Number of primes of the form 4k+1 less than 10^n.

1, 11, 80, 609, 4783, 39175, 332180, 2880504, 25423491, 227523275, 2059020280, 18803924340, 173032709183, 1602470783672, 14922284735484, 139619168787795, 1311778575685086, 12369977142579584, 117028833597800689, 1110409801150582707

Offset: 1

T. D. Noe, Dec 19 2003

nonn

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

Cf. A091099 (number of primes of the form 4k+3 less than 10^n).

cnt=0; k=0; Table[lim=10^n; While[4k+1

a(10)-a(16) from Robert G. Wilson v, Dec 22 2003

a(17)-a(20) from Marc Deleglise, Jun 28 2007

A091295 (Number of primes == 3 mod 4 less than 10^n) - (number of primes == 1 mod 4 less than 10^n).

Original entry on oeis.org

1, 2, 7, 10, 25, 147, 218, 446, 551, 5960, 14252, 63337, 118472, 183457, 951700, 3458334, 6284060, 2581691, 80743228, 259753425

Offset: 1

Enoch Haga, Feb 23 2004

			a(1) = 1 because below 10^1 3 and 7 are 3 mod 4 and 5 is 1 mod 4 and the difference is 2-1=1.

Hans Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Birkhauser, The distribution of primes between the two series 4n+1 and 4n+3, pages 73-77, with graphs.

Marc Deléglise, Pierre Dusart, and Xavier-Francois Roblot, Counting primes in residue classes, Math. Comp. 73 (2004), no. 247, 1565-1575.
Carlos Rivera, Puzzle 256, Jack Brennen old records, The Prime Puzzles & Problems Connection.

Cf. A091098, A091099. A002144, A002145, A091318, A091267.

a(n) = A091099(n) - A091098(n) = A093153(n) + 1. [Max Alekseyev, May 17 2009]

a(17)-a(20) from Marc Deleglise, Jun 28 2007

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)

A091100 Number of Gaussian primes whose norm is less than 10^n.

Original entry on oeis.org

16, 100, 668, 4928, 38404, 313752, 2658344, 23046512, 203394764, 1820205436, 16472216912, 150431552012, 1384262129028, 12819767598972, 119378281788240, 1116953361826164

Offset: 1

T. D. Noe, Dec 19 2003

nonn

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
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), A091101, A091102.
Cf. A091134 (number of Gaussian primes whose modulus is less than 10^n).
Cf. A017934, A295996.

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

a(2n) = 8*A091098(2n) + 4*A091099(n) + 4.

a(n) ~ 4 Li(10^n) ~ k/n * 10^n, where k = 4/log(10) = 1.737.... - Charles R Greathouse IV, Oct 24 2012

a(10)-a(16) from Seiichi Manyama using the data in A091098, Dec 03 2017

A091102 Number of first half-quadrant Gaussian primes whose norm is less than 10^n.

Original entry on oeis.org

3, 14, 87, 623, 4818, 39263, 332406, 2881124, 25425200, 227528084

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), A091101, A091102.

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

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

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

A093153 Difference between counts of odd composites in A093151 and A093152 [Count (1 mod 4) - count (3 mod 4)].

Original entry on oeis.org

0, 1, 6, 9, 24, 146, 217, 445, 550, 5959, 14251, 63336, 118471, 183456, 951699, 3458333, 6284059, 2581690, 80743227, 259753424

Offset: 1

Enoch Haga, Mar 24 2004

In A091295 the counts are 1 higher. I computed the differences through 10^8 and the rest by extrapolating from A091098 and A091099. In the ranges given, the counts of odd composites less than 10^n are higher 1 mod 4 than 3 mod 4. They are exactly opposite for the primes less than 10^n where 3 mod 4 is higher.

			Below 10^3 there are 169 odd composites 1 mod 4 and 163 odd composites 3 mod 4, so a(3)=169-163=6

Cf. A093151 A093152 A091295 A091098 A091099.

Subtract count of odd composites 3 mod 4 less than 10^n from those 1 mod 4

a(n) = A093151(n) - A093152(n). For n>1, a(n) = A091099(n) - A091098(n) - 1. [From Max Alekseyev, May 17 2009]

More terms from Max Alekseyev, May 17 2009

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

A093151 Number of odd composites 1 mod 4 less than 10^n.

Original entry on oeis.org

1, 13, 169, 1890, 20216, 210824, 2167819, 22119495, 224576508, 2272476724, 22940979719, 231196075659, 2326967290816, 23397529216327, 235077715264515, 2360380831212204, 23688221424314913, 237630022857420415, 2382971166402199310, 23889590198849417292

Offset: 1

Enoch Haga, Mar 24 2004

nonn

			Below 10^2 there are 13 odd composites so a(2)=13

Cf. A093152 A093153 A091295 A091098 A091099.

Count odd composites 1 mod 4 less than 10^n

For n>1, a(n) = 25*10^(n-2) - 1 - A091098(n). - Max Alekseyev, May 30 2007

More terms from Max Alekseyev, May 30 2007

Extended by Max Alekseyev, Oct 13 2009

A093152 Number of odd composites below 10^n that are congruent to 3 modulo 4.

Original entry on oeis.org

0, 12, 163, 1881, 20192, 210678, 2167602, 22119050, 224575958, 2272470765, 22940965468, 231196012323, 2326967172345, 23397529032871, 235077714312816, 2360380827753871, 23688221418030854, 237630022854838725, 2382971166321456083, 23889590198589663868

Offset: 1

Enoch Haga, Mar 24 2004

nonn

			Below 10^2 there are 12 odd composites 3 mod 4 so a(2)=12

Cf. A093151 A093153 A091295 A091098 A091099.

For n>1, a(n) = 25*10^(n-2) - A091099(n) [From Max Alekseyev, May 17 2009]

a(9) from Ryan Propper, Jun 21 2005

a(10)-a(20) from Max Alekseyev, May 17 2009, Oct 08 2013

cp's OEIS Frontend

A091098 Number of primes of the form 4k+1 less than 10^n.

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A091295 (Number of primes == 3 mod 4 less than 10^n) - (number of primes == 1 mod 4 less than 10^n).

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

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A091100 Number of Gaussian primes whose norm is less than 10^n.

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A091102 Number of first half-quadrant Gaussian primes whose norm is less than 10^n.

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A093153 Difference between counts of odd composites in A093151 and A093152 [Count (1 mod 4) - count (3 mod 4)].

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

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Mathematica

Formula

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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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Extensions

A093151 Number of odd composites 1 mod 4 less than 10^n.

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