A066520 -id:A066520 - cp's OEIS frontend

A051025 Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

26861, 616841, 616849, 616877, 617011, 617269, 617327, 617339, 617359, 617369, 617401, 617429, 617453, 617521, 617537, 617689, 617699, 617717, 622813, 622987, 623003, 623107, 623209, 623299, 623321, 623341, 623353, 623401, 623423, 623437

Offset: 1

Eric W. Weisstein

nonn

This is a companion sequence to A051024.
Starting from a(27556)=9103362505801 the sequence includes the 8th sign-changing zone predicted by C. Bays et al. The sequence with the first 8 sign-changing zones contains 418933 terms (see a-file) with a(418933)=9543313015309 as its last term. - Sergei D. Shchebetov, Oct 06 2017
We also discovered the 9th sign-changing zone, which starts from 64083080712569, ends with 64084318523021, and has 13370 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Dec 30 2017
We also discovered the 10th sign-changing zone, which starts from 715725135905981, ends with 732156384107921, and has 481194 terms with pi_{4,3}(p) - pi_{4,1}(p) = -1. This zone is considerably lower than predicted by M. Deléglise et al. in 2004. - Andrey S. Shchebetov and Sergei D. Shchebetov, Jan 28 2018

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; Journal of Algebra, Number Theory: Advances and Applications, 2013, 8 (1-2), pp.41-55.
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.
Sergei D. Shchebetov, First 418933 terms (zipped file)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

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

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). - Daniel Forgues, Mar 26 2009

For[i=2; d=0, True, i++, d+=Mod[p=Prime[i], 4]-2; If[d==-1, Print[p]]]
(* Second program: *)
Prime@ Position[Accumulate@ Array[Mod[Prime@ #, 4] - 2 &, 51000], -1][[All, 1]] (* Michael De Vlieger, Dec 30 2017 *)

from sympy import nextprime; a, p = 0, 2
while p < 623803:
    p=nextprime(p); a += p%4-2
    if a == -1: print(p, end = ', ')  # Ya-Ping Lu, Jan 18 2025

Edited by Dean Hickerson, Mar 10 2002

A156749 For all numbers k(n) congruent to -1 or +1 (mod 4) starting with k(n) = {3,5,7,9,11,...}, a(k(n)) is incremented by the congruence (mod 4) if k(n) is prime and by 0 if k(n) is composite.

Original entry on oeis.org

-1, 0, -1, -1, -2, -1, -1, 0, -1, -1, -2, -2, -2, -1, -2, -2, -2, -1, -1, 0, -1, -1, -2, -2, -2, -1, -1, -1, -2, -1, -1, -1, -2, -2, -3, -2, -2, -2, -3, -3, -4, -4, -4, -3, -3, -3, -3, -2, -2, -1, -2, -2, -3, -2, -2, -1, -1, -1, -1, -1, -1, -1, -2, -2, -3, -3, -3, -2, -3, -3

Offset: 1

Daniel Forgues, Feb 14 2009

sign

The fact that a(k(n)) is predominantly negative exhibits the Chebyshev Bias (where the congruences that are not quadratic residues generally lead in the prime number races, at least for "small" integers, over the congruences that are quadratic residues).
This bias seems caused (among other causes?) by the presence of all those squares (even powers) coprime to 4 taking away opportunities for primes to appear in the quadratic residue class +1 (mod 4), while the non-quadratic residue class -1 (mod 4) is squarefree.
The density of squares congruent to +1 (mod 4) is 1/(4*sqrt(k(n))) since 1/2 of squares are congruent to +1 (mod 4), while the density of primes in either residue class -1 or +1 (mod 4) is 1/(phi(4)*log(k(n))), with phi(4) = 2.
Here 1 is quadratic residue mod 4, but 3 (or equivalently -1) is quadratic non-residue mod 4. All the even powers (included in the squares) map congruences {-1, +1} to {+1, +1} respectively and so contribute to the bias, whereas all the odd powers map {-1, +1} to {-1, +1} respectively and so do not contribute to the bias.
One would then expect the ratio of this bias, if caused exclusively by the even powers, relative to the number of primes in either congruences to asymptotically tend towards to 0 as k(n) increases (since 1/(4*sqrt(k(n))) is o(1/(phi(4)*log(k(n))))).
The persistence or not of such bias in absolute value then does not contradict The Prime Number Theorem for Arithmetic Progressions (Dirichlet) which states that the asymptotic (relative) ratio of the count of prime numbers in each congruence class coprime to m tends to 1 in the limit towards infinity. (Cf. 'Prime Number Races' link below.)
Also, even if this bias grows in absolute value, it is expected to be drowned out (albeit very slowly) by the increasing fluctuations in the number of primes in each congruence class coprime to 4 since, assuming the truth of the Riemann Hypothesis, their maximum amplitude would be, with x standing for k(n) in our case, h(x) = O(sqrt(x)*log(x)) <= C*sqrt(x)*log(x) in absolute value which gives relative fluctuations of order h(x)/x = O(log(x)/sqrt(x)) <= C*log(x)/sqrt(x) in the densities of primes pi(x, {4, 1})/x and pi(x, {4, 3})/x in either congruence class.
Since 1/(4*sqrt(x)) is o(log(x)/sqrt(x)) the bias will eventually be overwhelmed by the "pink noise or nearly 1/f noise" corresponding to the fluctuations in the prime densities in either congruence class. The falsehood of the Riemann Hypothesis would imply even greater fluctuations since the RH corresponds to the minimal h(x).
We get pink noise or nearly 1/f noise if we consider the prime density fluctuations of pi(x, {4, k})/x as an amplitude spectrum over x (with a power density spectrum of (C*log(x)/sqrt(x))^2 = ((C*log(x))^2)/x and see x as the frequency f. This power density spectrum is then nearly 1/x and would have nearly equal energy (although slowly increasing as (C*log(x))^2) for each octave of x. (Cf. 'Prime Numbers: A Computational Perspective' link below.)
Among the positive integers k(n) up to 100000 that are congruent to -1 or +1 (mod 4) [indexed from n = 1 to 49999, with k(n) = 4*ceiling(n/2) + (-1)^n], a tie is attained or maintained, with a(k(n)) = 0, for only 34 integers and that bias favoring the non-quadratic residue class -1 (mod 4) gets violated only once, i.e., a(k(n)) = +1, for index n = 13430 (corresponding to the prime k(n) = 26861 congruent to +1 (mod 4) since n is even) where the congruence +1 leads once!

Richard E. Crandall and Carl Pomerance, Prime Numbers: A Computational Perspective

Daniel Forgues, Table of n, a(n) for n = 1..49999
A. Granville and G. Martin, Prime Number Races, arXiv:math/0408319 [math.NT], 2004.
Eric Weisstein, Chebyshev Bias
Wikipedia, Pink noise

Cf. A066339, A066490, A066520, A007350, A007351, A038691, A096628, A156707, A156709, A156706, A101264, A075743.

Table[Which[!PrimeQ[2*n+1], 0, Mod[2*n+1, 4] == 1, 1, True, -1], {n, 1, 100}] // Accumulate (* Jean-François Alcover, Dec 09 2014 *)

a(n) = -A066520(2*n+1) = A066339(2*n+1) - A066490(2*n+1). - Jonathan Sondow, May 17 2013

Edited by Daniel Forgues, Mar 01 2009, Mar 29 2009

A151763 If n is a prime == 1 mod 4 then a(n) = 1, if n is a prime == 3 mod 4 then a(n) = -1, otherwise a(n) = 0.

Original entry on oeis.org

0, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0

Offset: 1

N. J. A. Sloane, Jun 22 2009

sign

a(A002145(n)) = -1; a(A065090(n)) = 0; a(A002144(n)) = 1. [Reinhard Zumkeller, Oct 06 2011]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Katz, Lang-Trotter revisited, Bull. Amer. Math. Soc., 46 (2009), 413-457.

Cf. A079260, A079261.

Cf. A066520 (partial sums).

a151763 n | even n         = 0
          | a010051 n == 1 = 2 - n `mod` 4
          | otherwise      = 0
-- Reinhard Zumkeller, Oct 06 2011

a:= proc(n) if n::odd and isprime(n) then 2 - (n mod 4) else 0 fi end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 22 2014

a[n_] := Which[!PrimeQ[n], 0, m = Mod[n, 4]; m == 1, 1, m == 3, -1, True, 0]; Array[a, 105] (* Jean-François Alcover, Dec 03 2016 *)

a(n) = (2 - n mod 4) * A010051(n).

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).

Original entry on oeis.org

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

A348446 a(n) = A307720(2n-1) - A307220(2n).

Original entry on oeis.org

0, 1, 2, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 4, 4, 4, 6, 6, 6, 5, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 4, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 1, 1, 1, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Oct 22 2021

sign

My guess is that this changes signs infinitely often, but is more likely to be positive than negative. Perhaps the behavior is akin to that of A066520, which shows the "great prime race" between primes congruent to 3 mod 4 and primes congruent to 1 mod 4.

See also the graphs in A307720 and A348248.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Rémy Sigrist, Scatterplot of the first million terms
Rémy Sigrist, Scatterplot of the first five million terms
N. J. A. Sloane, Table of n, a(n) for n = 1..500000
Chai Wah Wu, Scatterplot of the first 100 million terms

Cf. A066520, A307720, A307730, A348248.

from itertools import islice
from collections import Counter
def A348446(): # generator of terms. Greedy algorithm
    a = 1
    c, b = Counter(), 1
    while True:
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        a2 = k
        yield a-a2
        k, kb = 1, b
        while c[kb] >= kb:
            k += 1
            kb += b
        c[kb] += 1
        b = k
        a = k
A348446_list = list(islice(A348446(),100)) # Chai Wah Wu, Oct 23 2021

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.

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.

A329243 a(n) = Pi(8,3)(prime(n)) + Pi(8,5)(prime(n)) + Pi(8,7)(prime(n)) - 3Pi(8,1)(prime(n)), where Pi(a,b)(x) denotes the number of primes in the arithmetic progression ak + b less than or equal to x.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 11, 8, 9, 10, 7, 4, 5, 6, 7, 8, 5, 6, 7, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 16, 17, 14, 15, 12, 13, 10, 11, 12, 13, 14, 11, 12, 13, 14, 15, 12, 13, 14, 11, 12, 13, 10, 11, 12, 13, 14, 15, 16

Offset: 1

Jianing Song, Nov 08 2019

sign

The initial terms are nonnegative integers, a(n) is negative for some prime(n) ~ 10^28.127. See the comments about "Chebyshev's bias" in A329242.

			For prime(25) = 97, there are 5 primes <= 97 that are congruent to 1 mod 8 (17, 41, 73, 89, 97), 7 primes congruent to 3 mod 8 (3, 11, 19, 43, 59, 67, 83), 6 primes congruent to 5 mod 8 (5, 13, 29, 37, 53, 61), 6 primes congruent to 7 mod 8 (7, 23, 31, 47, 71, 79), so a(25) = 7 + 6 + 6 - 3*5 = 4.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A329242.

Cf. A321856, A066520, A321857, A321859, A321860.

a(n) = my(k=0); forprime(p=3, prime(n), if(p%8==1, k-=3, k++)); k

Edited by Peter Munn, Nov 19 2023

cp's OEIS Frontend

A051025 Primes p for which pi_{4,3}(p) - pi_{4,1}(p) = -1, where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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A156749 For all numbers k(n) congruent to -1 or +1 (mod 4) starting with k(n) = {3,5,7,9,11,...}, a(k(n)) is incremented by the congruence (mod 4) if k(n) is prime and by 0 if k(n) is composite.

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A151763 If n is a prime == 1 mod 4 then a(n) = 1, if n is a prime == 3 mod 4 then a(n) = -1, otherwise a(n) = 0.

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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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Python

A348446 a(n) = A307720(2*n-1) - A307220(2*n).

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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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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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A348446 a(n) = A307720(2n-1) - A307220(2n).

A329243 a(n) = Pi(8,3)(prime(n)) + Pi(8,5)(prime(n)) + Pi(8,7)(prime(n)) - 3Pi(8,1)(prime(n)), where Pi(a,b)(x) denotes the number of primes in the arithmetic progression ak + b less than or equal to x.