A096452 -id:A096452 - 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

A096629 Values of n for which {p_3, p_4, ..., p_n} (mod 3) contains equal numbers of 1's and 2's.

Original entry on oeis.org

4, 6, 8, 12, 14, 22, 38, 48, 50, 23338590786, 23338590788, 23338590790, 23338590806, 23338590808, 23338590820, 23338590822, 23338590824, 23338590826, 23338590828, 23338590830, 23338590834, 23338590840, 23338590858, 23338590860, 23338590868, 23338590870

Offset: 1

Eric W. Weisstein, Jul 04 2004

nonn

Donovan Johnson, Table of n, a(n) for n = 1..85508
C. Bays and R. H. Hudson, Details of the first region of integers x with pi_{3,2}(x) < pi_{3,1}(x), Math. Comp. 32 (1978), 571-576.
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A000720, A096630, A096449, A098044, A096452.

a(n) = A000720(A098044(n+1)).

Edited by Max Alekseyev, Sep 13 2009

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

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

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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A096629 Values of n for which {p_3, p_4, ..., p_n} (mod 3) contains equal numbers of 1's and 2's.

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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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