A096449 -id:A096449 - 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

A297006 Primes p for which pi_{3,2}(p) - pi_{3,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

608981813029, 608981813137, 608981813261, 608981813273, 608981813311, 608981813357, 608981813459, 608981813683, 608981813717, 608981813777, 608981813789, 608981814127, 608981818999, 608981819021, 608981819273, 608981819359, 608981819419, 608981820869, 608981820899, 608981820913, 608981826877, 608981827873, 608981827891, 608981828023, 608981828029, 608981828111, 608981828129, 608981836363, 608981836391, 608981836481

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 23 2017

nonn

This sequence is a companion sequence to A297005. Starting from a(20591)=6148171711663 the sequence includes the second sign-changing zone predicted by C. Bays et al. in 2001. The sequence with the first two sign-changing zones up to 10^13 contains 84323 terms with a(84323)=6156051951677 as its last term (see b-file). In addition, a(1) = A007352(2) as well as a(20591) = A007352(9630).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..84323
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
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
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp. 54-76.
M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 1565-1575.
A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
M. Rubinstein and P. Sarnak, Chebyshev’s bias, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007352, A096629, A096630, A096449, A098044.

A297005 Values of n for which pi_{3,2}(p_n) - pi_{3,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

Original entry on oeis.org

23338590792, 23338590794, 23338590796, 23338590798, 23338590800, 23338590802, 23338590804, 23338590810, 23338590814, 23338590816, 23338590818, 23338590832, 23338591016, 23338591018, 23338591028, 23338591030, 23338591032, 23338591084, 23338591086, 23338591088, 23338591302, 23338591340, 23338591342, 23338591344, 23338591346, 23338591348, 23338591350, 23338591656, 23338591658, 23338591662

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 23 2017

nonn

This sequence is a companion sequence to A297006. Starting from a(20591)=216415270060 the sequence includes the second sign-changing zone predicted by C. Bays et al. in 2001. The sequence with the first two sign-changing zones up to 10^13 contains 84323 terms with a(84323)=216682882512 as its last term (see b-file). In addition, a(1) = A096630(1).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..84323
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
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.
C. Bays, K. Ford, R. H. Hudson and M. Rubinstein, Zeros of Dirichlet L-functions near the real axis and Chebyshev's bias, J. Number Theory 87 (2001), pp. 54-76.
M. Deléglise, P. Dusart, and X. Roblot, Counting Primes in Residue Classes, Mathematics of Computation, American Mathematical Society, 2004, 73 (247), pp. 1565-1575.
A. Granville and G. Martin, Prime Number Races, Amer. Math. Monthly 113 (2006), no. 1, 1-33.
M. Rubinstein and P. Sarnak, Chebyshev's bias, Experimental Mathematics, Volume 3, Issue 3, 1994, pp. 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007352, A096629, A096630, A096449, A098044.

A174695 Partial sums of A173950.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 0, 1, 2, 3, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2

Offset: 1

Giovanni Teofilatto, Nov 30 2010

sign

Since this sequence equals A112632(n)-1, and A007352 gives the primes at which the sign of A112632 changes, we have a change of sign in the present sequence not exactly at the primes listed in A007352, but earlier for changes to negative sign, and later for the opposite changes. Moreover, a change of sign in either of the sequences corresponds not necessarily to a change of sign (in the strict sense, i.e., regarding 0 as a number with the same sign as the preceding term) in the other one. - M. F. Hasler, Oct 09 2011

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10000

Cf. A007352, A112632, A098044, A096630.

Concerning zeros or changes of sign, see also A096449 and A275939.

A173950 := proc(n) if (ithprime(n)+1) mod 6 = 0 then 1; elif (ithprime(n)-1) mod 6 = 0 then -1; else 0 ; end if; end proc:
A174695 := proc(n) add(A173950(i),i=1..n) ; end proc:
seq(A174695(n),n=1..90) ; # R. J. Mathar, Nov 30 2010

Accumulate[Table[Which[Divisible[Prime[n]+1,6],1,Divisible[Prime[n]-1,6],-1,True,0],{n,150}]] (* Harvey P. Dale, Apr 24 2019 *)

s=0;forprime(p=1,999,print1(s+=if(p%3-1,p>3,-1)","))  \\ M. F. Hasler, Oct 09 2011

a(n) = A112632(n)-1. - M. F. Hasler, Oct 09 2011

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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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A297006 Primes p for which pi_{3,2}(p) - pi_{3,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A297005 Values of n for which pi_{3,2}(p_n) - pi_{3,1}(p_n) = -1, where p_n is the n-th prime and pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A174695 Partial sums of A173950.

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