A007351 -id:A007351 - cp's OEIS frontend

A297447 Values of n for which pi_{8,5}(p_n) - pi_{8,1}(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).

30733704, 30733708, 30733714, 30733726, 30733729, 30733733, 30733743, 30733762, 30733764, 30733777, 30733781, 30733796, 30733853, 30733857, 30733860, 30733866, 30733880, 30733887, 30733890, 30734262

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017

nonn

This is a companion sequence to A297448. The first two sign-changing zones were discovered by Bays and Hudson back in 1979. We discovered four additional zones starting from a(22794) = 186422420112. The full sequence with all 6 zones checked up to 5*10^14 contains 664175 terms (see a-file) with a(664175) = 6097827689926 as its last term.
This sequence was checked up to 10^15 and the 7th sign-changing zone starting from a(664176) = 27830993289634 and ending with a(850232)= 27876113171315 was found. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jul 28 2018
The y-coordinate of prime(a(n)) on the Cartesian grid defined in A379643 is -1. - Ya-Ping Lu, Jan 08 2025

Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
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, Pages 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354, A379643, A379989.

from sympy import nextprime; p, r1, r5 = 1, 0, 0
for n in range(1, 30734263):
    p = nextprime(p); r = p%8
    if r == 1: r1 += 1
    elif r == 5: r5 += 1
if r in {1, 5} and r1 == r5 + 1: print(n, end = ', ')  # Ya-Ping Lu, Jan 08 2025

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

A295354 Primes p for which pi_{8,7}(p) - pi_{8,1}(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

192252423729713, 192252423730849, 192252423731231, 192252423731633, 192252423731663, 192252423731839, 192252423732311, 192252423769201, 192252423769361, 192252423769537, 192252423772649, 192252423772807, 192252423772847, 192252423774023, 192252423774079, 192252423774457, 192252423779257, 192252423782521, 192252423783263, 192252423783551

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Nov 20 2017

nonn

This sequence is a companion sequence to A295353. The sequence with the first found pi_{8,7}(p_n) - pi_{8,1}(p_n) sign-changing zone contains 234937 terms (see a-file) with a(237937) = 192876135747311 as its last term. In addition, a(1) = A275939(8).

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
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, Pages 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354.

A093180 Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

Original entry on oeis.org

26829, 26845, 26853, 26857, 26865, 26869, 26873, 26877, 26885, 26889, 26897, 26917, 616765, 616773, 616777, 616785, 616825, 616833, 616837, 616845, 616853, 616857, 616861, 616865, 616869, 616873, 616881, 616885, 616889, 616893, 617013

Offset: 1

Enoch Haga, Mar 27 2004

This odd composite race seems analogous to the prime number race

			a(1)=26829 is the first odd composite 1 mod 4 where the count 5238 is the same for 26835 3 mod 4

Cf. A007350, A007351, A038691, A093181, A093182.

Beginning with 1, run separate counts for odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the number for 1 mod 4.

A093181 Odd composites (including 1 in the count) where the number 3 mod 4 equals the number 1 mod 4.

Original entry on oeis.org

26835, 26851, 26855, 26859, 26867, 26871, 26875, 26883, 26887, 26895, 26899, 26923, 616771, 616775, 616779, 616795, 616831, 616835, 616839, 616847, 616851, 616855, 616859, 616863, 616867, 616875, 616879, 616883, 616887, 616891, 617007

Offset: 1

Enoch Haga, Mar 27 2004

This odd composite race seems analogous to the prime number race

			a(1)=26835 is the first odd composite 3 mod 4 where the count 5238 is the same for 26829 1 mod 4

Cf. A007350, A007351, A038691, A093180, A093182.

Beginning with 1, run separate counts for odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the number for 3 mod 4.

A093182 Counts where both the odd composites (starting from 1) 1 mod 4 and 3 mod 4 are equal.

Original entry on oeis.org

5238, 5241, 5242, 5243, 5244, 5245, 5246, 5247, 5248, 5249, 5250, 5255, 129008, 129009, 129010, 129012, 129020, 129021, 129022, 129023, 129024, 129025, 129026, 129027, 129028, 129029, 129030, 129031, 129032, 129033, 129058, 129059, 129060

Offset: 1

Enoch Haga, Mar 27 2004

			At 26829 1 mod 4 and 26835 3 mod 4, the count of odd composites is equal for each run at 5238; so a(1)=5238. [Compare to the prime 26833 1 mod 4 where equality occurs at count 1471 and the first reversal in the race occurs at 26861.]

Cf. A093180, A093181, A007350, A007351, A038691.

Run separate counts of odd composites 1 mod 4 and 3 mod 4. When the count is equal, record the count.

A216057 a(n) = A045429(n) - A045356(n).

Original entry on oeis.org

1, 6, 6, 6, 12, 12, 6, 6, 2, 4, -6, -6, 4, -4, 4, 4, 14, 24, 24, 24, 24, 14, 14, 24, 6, 12, 12, 6, 2, 4, -24, -24, -26, -34, -18, -6, 6, 4, 12, 24, 22, 14, 4, 12, 6, 24, 24, 34, 24, 32, 16, 14, 24, 24, 26, 32, 34, 26, 34, 14, 14, 6, -18, -18, 6, 4, -6, -8, -14

Offset: 1

Zak Seidov, Aug 31 2012

sign

Zak Seidov, Table of n, a(n) for n = 1..1000
Zak Seidov, Plot of terms up to a(1000) (gif image)
A. Granville and G. Martin, Prime number races, arXiv:math/0408319v1 [math.NT]
Eric Weisstein's World of Mathematics, Chebyshev Bias

Cf. A007350, A007351, A038691, A038698, A045356, A045429, A156749.

A295353 Values of n for which pi_{8,7}(p_n) - pi_{8,1}(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

6035005477560, 6035005477596, 6035005477608, 6035005477618, 6035005477620, 6035005477623, 6035005477632, 6035005478719, 6035005478725, 6035005478730, 6035005478822, 6035005478826, 6035005478829, 6035005478863, 6035005478866, 6035005478874, 6035005479026, 6035005479132, 6035005479158, 6035005479163

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Nov 20 2017

nonn

This sequence is a companion sequence to A295354. The sequence with the first found pi_{8,7}(p_n) - pi_{8,1}(p_n) sign-changing zone contains 234937 terms (see a-file) with a(237937) = 6053968231350 as its last term.

Andrey S. Shchebetov and Sergei D. Shchebetov, Table of n, a(n) for n = 1..100000
A. Alahmadi, M. Planat, and P. Solé, Chebyshev's bias and generalized Riemann hypothesis, HAL Id: hal-00650320.
C. Bays and R. H. Hudson, Numerical and graphical description of all axis crossing regions for moduli 4 and 8 which occur before 10^12, International Journal of Mathematics and Mathematical Sciences, vol. 2, no. 1, pp. 111-119, 1979.
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, Pages 173-197.
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A007350, A007351, A038691, A051024, A051025, A066520, A096628, A096447, A096448, A199547, A275939, A295354

A297354 Values of n for which pi_{12,5}(p_n) - pi_{12,1}(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

862062606318, 862062606330, 862062606348, 862062606351, 862062606377, 862062606380, 862062606387, 862062606393, 862062606424, 862062606448, 862062606453, 862062606466, 862062606469, 862062606478, 862062606481, 862062606488, 862062606490, 862062606494, 862062606496, 862062606500

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 29 2017

nonn

This is a companion sequence to A297355 and includes values of n for the first discovered sign-changing zone for pi_{12,5}(p) - pi_{12,1}(p) prime race. The full sequence checked up to 10^14 has 8399 terms (see b-file).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..8399
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.
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. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A297355.

A297355 Primes p for which pi_{12,5}(p) - pi_{12,1}(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

25726067172577, 25726067172857, 25726067173321, 25726067173441, 25726067174389, 25726067174461, 25726067174653, 25726067174761, 25726067175961, 25726067176549, 25726067176669, 25726067176993, 25726067177149, 25726067177429, 25726067177449, 25726067177593, 25726067177617, 25726067177689, 25726067177801, 25726067178013

Offset: 1

Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 29 2017

nonn

This is a companion sequence to A297354 and includes the first discovered sign-changing zone for pi_{12,5}(p) - pi_{12,1}(p) prime race. The full sequence checked up to 10^14 has 8399 terms (see b-file).

Sergei D. Shchebetov, Table of n, a(n) for n = 1..8399
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.
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. A007350, A007351, A038691, A051024, A066520, A096628, A096447, A096448, A199547, A297354.

cp's OEIS Frontend

A297447 Values of n for which pi_{8,5}(p_n) - pi_{8,1}(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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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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A295354 Primes p for which pi_{8,7}(p) - pi_{8,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A093180 Odd composites (including 1 in the count) where the number 1 mod 4 equals the number 3 mod 4.

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A093181 Odd composites (including 1 in the count) where the number 3 mod 4 equals the number 1 mod 4.

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A093182 Counts where both the odd composites (starting from 1) 1 mod 4 and 3 mod 4 are equal.

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A216057 a(n) = A045429(n) - A045356(n).

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A295353 Values of n for which pi_{8,7}(p_n) - pi_{8,1}(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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A297354 Values of n for which pi_{12,5}(p_n) - pi_{12,1}(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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A297355 Primes p for which pi_{12,5}(p) - pi_{12,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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