A096455 -id:A096455 - cp's OEIS frontend

A096447 Odd primes p such that the number of primes less than p that are congruent to 1 (mod 4) is equal to the number of primes less than p that are congruent to 3 (mod 4).

3, 7, 19, 43, 463, 26839, 26861, 26879, 26891, 26903, 26927, 616783, 616799, 616841, 616849, 616877, 617039, 617269, 617369, 617401, 617429, 617453, 617471, 617479, 617521, 617537, 617587, 617689, 617717, 617723, 618439, 618547, 618619, 618643

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

Assign the odd prime numbers to the rows of an array as follows:
Assign the first odd prime, prime(2) = 3, to row 0 (the top row).
For m > 2, assign prime(m) to the row immediately above or below the row to which prime(m-1) was assigned: above if prime(m-1) == 1 (mod 4), below otherwise.
The following array results:
  row 0 (this sequence): 3, 7, 19, 43, 463, 26839, ...
  row 1 (A096448):       5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127, ...
  row 2 (A096451):       13, 29, 37, 53, 61, 71, 79, 101, 107, 113 ...
  row 3:                 73, 83, 97, 109, ...
  row 4:                 89, ...

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A007351, A151800.

Cf. A096448, A096449, A096450, A096451, A096452, A096453, A096454, A096455.

lim = 10^5; k1 = 0; k3 = 0; p = 2; t = {}; Do[p = NextPrime[p]; If[k1 == k3, AppendTo[t, p]]; If[Mod[p, 4] == 1, k1++, k3++], {lim}]; t (* T. D. Noe, Sep 07 2011 *)

a(n) = A151800(A007351(n)), the next prime after A007351(n). - Joshua Zucker, May 03 2006

More terms from Joshua Zucker, May 03 2006

"odd" added to definition by N. J. A. Sloane, Sep 09 2015

A096448 Primes p such that the number of primes less than p equal to 1 mod 4 is one less than the number of primes less than p equal to 3 mod 4.

Original entry on oeis.org

5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127, 419, 431, 439, 461, 467, 1259, 1279, 1303, 26833, 26849, 26881, 26893, 26921, 26947, 615883, 616769, 616787, 616793, 616829, 617051, 617059, 617087, 617257, 617473, 617509, 617647, 617681, 617731, 617819, 617879

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

			First term prime(2) = 3 is placed on 0th row.
If prime(n-1) = +1 mod 4 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 4 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
  0th row:  3,  7, 19,  43, ...
  1st row:  5, 11, 17,  23, 31, 41, 47, 59, 67, 103, 127, ...
  2nd row: 13, 29, 37,  53, 61, 71, 79, 101, 107, 113 ...
  3rd row: 73, 83, 97, 109, ...
  4th row: 89, ...

Michel Marcus, Table of n, a(n) for n = 1..751

Cf. A096447, A096449, A096450, A096451, A096452, A096453, A096454, A096455.

Prime[#]&/@(Flatten[Position[Accumulate[If[Mod[#,4]==1,1,-1]&/@ Prime[ Range[ 2,51000]]],-1]]+2) (* Harvey P. Dale, Mar 08 2015 *)

lista(nn) = my(vp=primes(nn), nb1=0, nb3=0); for (i=2, #vp, my(p = vp[i]); if (nb1 == nb3-1, print1(p, ", ")); if ((p % 4) == 1, nb1++, nb3++);); \\ Michel Marcus, May 30 2024

More terms from Joshua Zucker, May 03 2006

A096449 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is equal to the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

5, 11, 17, 23, 41, 47, 83, 167, 227, 233, 608981812919, 608981812961, 608981813017, 608981813569, 608981813677, 608981813833, 608981813851, 608981813927, 608981813939, 608981813963, 608981814043, 608981814149, 608981814251, 608981814827

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

First term prime(3) = 5 is placed on 0th row.
If prime(n-1) = +1 mod 3 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 3 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
5, 11, 17, 23, 41, 47, 83, ... (this sequence)
7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101, .. (A096452).
31, 59, 67, 73, 97, ... (A096453)
61, ...

Cf. A096448-A096455.

lst = {5}; p = 0; q = 0; r = 5; While[r < 10^9, If[ Mod[r, 3] == 2, p++, q++ ]; r = NextPrime@r; If[p == q, AppendTo[lst, r]; Print@r]]; lst (* Robert G. Wilson v, Sep 20 2009 *)

For n>1, a(n) = prime(A096629(n-1)+1) = A000040(A096629(n-1)+1). - Max Alekseyev, Sep 19 2009

a(n) = A151800(A098044(n)) = A007918(A098044(n)+1).

More terms and better definition from Joshua Zucker, May 21 2006

Terms a(11) onward from Max Alekseyev, Feb 10 2011

A096452 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is one less than the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

7, 13, 19, 29, 37, 43, 53, 71, 79, 89, 101, 107, 113, 131, 163, 173, 223, 229, 239, 251, 383, 443, 1811, 1871, 1877, 1889, 608981812613, 608981812667, 608981812891, 608981812951, 608981812993, 608981813929, 608981813941, 608981814019, 608981814173

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

A. Granville, G. Martin, Prime number races, Amer. Math. Monthly vol 113, no 1 (2006) p 1.
G. D. Martin, Biases in the Shanks-Renyi prime number race, FoCM'02 Conference, Minneapolis, 5-14 Aug 2002.

Cf. A096447-A096455, A096629, A098044.

lst = {}; p = q = 0; r = 5; While[r < 10^5, If[ Mod[r, 3] == 2, p++, q++ ]; r = NextPrime@ r; If[p == q + 1, AppendTo[lst, r]; Print@ r]]; lst (* Robert G. Wilson v, Sep 20 2009 *)

More terms and better definition from Joshua Zucker, May 21 2006

Terms a(27) onward from Max Alekseyev, Feb 10 2011

A096453 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is two less than the number of such primes congruent to 2 mod 3.

Original entry on oeis.org

31, 59, 67, 73, 97, 103, 109, 127, 137, 149, 157, 179, 191, 197, 211, 241, 257, 347, 353, 379, 389, 401, 419, 431, 439, 449, 461, 467, 761, 773, 797, 1787, 1801, 1823, 1847, 1867, 1873, 1879, 1901, 3761, 9203, 198479, 198593, 608981812531, 608981812651, 608981812697

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A096447-A096455.

ct := -2: q:=2: for n from 2 to 20000 do p:=q: q:=ithprime(n): ct:=ct+`if`(p mod 3 = 1, -1, 1): if(ct=2)then printf("%d, ", q): fi: od: # Nathaniel Johnston, Jun 16 2011

npc1Q[n_]:=Module[{tot=PrimePi[n]-3,c},c=Count[Prime[Range[3,tot+2]],?(Mod[ #,3]==1&)];c==tot/2-1]; Select[Prime[Range[18000]],npc1Q] (* _Harvey P. Dale, Aug 22 2013 *)

lista(limit=10^6)={my(c=2); forprime(p=5, limit, if(!c, print1(p, ", ")); if(p%3==1, c++, c--))} \\ Andrew Howroyd, Aug 09 2025

More terms and better definition from Joshua Zucker, May 21 2006

a(44) onwards from Andrew Howroyd, Aug 09 2025

A096454 Primes p such that the number of primes q, 7 <= q < p, congruent to 1 or 2 mod 5, is one more than the number of such primes congruent to 3 or 4 mod 5.

Original entry on oeis.org

11, 17, 23, 41, 47, 59, 67, 83, 103, 109, 137, 149, 157, 167, 179, 191, 197, 211, 227, 233, 257, 269, 277, 307, 389, 401, 419, 431, 439, 499, 563, 587, 599, 607, 617, 631, 661, 677, 691, 727, 739, 761, 773, 797, 811, 853, 859, 883, 907, 937, 1171, 1289, 1297

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

Cf. A096447-A096455.

More terms and better definition from Joshua Zucker, May 21 2006

A096451 Primes p such that the number of primes less than p equal to 1 mod 4 is two less than the number of primes less than p equal to 3 mod 4.

Original entry on oeis.org

13, 29, 37, 53, 61, 71, 79, 101, 107, 113, 131, 139, 151, 163, 199, 359, 409, 421, 433, 443, 457, 479, 1223, 1231, 1249, 1277, 1283, 1291, 1301, 1307, 1399, 1423, 1439, 8699, 8779, 26821, 26951, 26959, 26987, 27011, 27031, 615731, 615869, 615887

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

nonn

First term prime(2) = 3 is placed on 0th row.
If prime(n-1) = +1 mod 4 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 mod 4 is on k-th row then we put prime(n) on (k+1)-st row.
This process makes an array of prime numbers:
3, 7, 19, 43, ....0th row
5, 11, 17, 23, 31, 41, 47, 59, 67, 103, 127, ....first row
13, 29, 37, 53, 61, 71, 79, 101, 107, 113 ....2nd row
73, 83, 97, 109, ....3rd row
89, ....4th row

Robert Israel, Table of n, a(n) for n = 1..811
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

Cf. A096447-A096455.

Cf. A002144, A002145, A007350, A007351

c1:= 0; c3:= 0: p:= 2: count:= 0: Res:= NULL:
while count < 100 do
  p:= nextprime(p);
  if c1 = c3 - 2 then
    count:= count+1;
    Res:= Res, p;
  fi;
  if p mod 4 = 1 then c1:=c1+1
  else c3:= c3+1
  fi
od:
Res; # Robert Israel, Nov 07 2018

More terms from Joshua Zucker, May 03 2006

A096450 Primes p such that the number of primes q, 7 <= q < p, congruent to 1 or 2 mod 5, is equal to the number of such primes congruent to 3 or 4 mod 5.

Original entry on oeis.org

7, 29, 37, 61, 89, 101, 107, 113, 131, 151, 181, 239, 251, 271, 397, 421, 443, 463, 479, 491, 503, 557, 569, 577, 601, 743, 757, 787, 857, 863, 881, 887, 1291, 1511, 1531, 1549, 1609, 1657, 1667, 1693, 1699, 1861, 1987, 1997, 2003, 2017, 2053, 2377, 2393

Offset: 1

Yasutoshi Kohmoto, Aug 12 2004

First term prime(4) = 7 is placed on 0th row.
If prime(n-1) = 1 or 2 mod 5 is on k-th row then we put prime(n) on (k-1)-st row.
If prime(n-1) = -1 or -2 mod 5 is on k-th row then we put prime(n) on (k+1)-st row.
This process produces the following array of prime numbers:
31, 97, ... row -1
7, 29, 37, 61, 89, 101, ... row 0 (this sequence)
11, 17, 23, 41, 47, 59, 67, 83, ... row 1 (A096454)
13, 19, 43, 53, 71, 79, ... row 2 (A096455)
73, ...

Cf. A096448-A096455.

More terms and better definition from Joshua Zucker, May 21 2006

cp's OEIS Frontend

A096447 Odd primes p such that the number of primes less than p that are congruent to 1 (mod 4) is equal to the number of primes less than p that are congruent to 3 (mod 4).

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A096448 Primes p such that the number of primes less than p equal to 1 mod 4 is one less than the number of primes less than p equal to 3 mod 4.

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A096449 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is equal to the number of such primes congruent to 2 mod 3.

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A096452 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is one less than the number of such primes congruent to 2 mod 3.

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A096453 Primes p such that the number of primes q, 5 <= q < p, congruent to 1 mod 3, is two less than the number of such primes congruent to 2 mod 3.

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A096454 Primes p such that the number of primes q, 7 <= q < p, congruent to 1 or 2 mod 5, is one more than the number of such primes congruent to 3 or 4 mod 5.

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A096451 Primes p such that the number of primes less than p equal to 1 mod 4 is two less than the number of primes less than p equal to 3 mod 4.

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Author

Keywords

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Crossrefs

Programs

Maple

Extensions

A096450 Primes p such that the number of primes q, 7 <= q < p, congruent to 1 or 2 mod 5, is equal to the number of such primes congruent to 3 or 4 mod 5.

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Author

Keywords

Comments

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