A007350 -id:A007350 - cp's OEIS frontend

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).

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.

A329224 a(n) is the smallest prime q such that Sum_{primes r <= q} Kronecker(r,prime(n)) > 0 (or equivalently, Sum_{primes r <= q} Kronecker(r,prime(n)) = 1), or 0 if no such prime exists.

Original entry on oeis.org

11100143, 608981813029, 2082927221, 2, 5, 2083, 2, 11, 2, 719, 2, 11, 2, 53, 2, 17, 5, 5, 163, 2, 2, 2, 11, 2, 2, 23, 2, 23, 5, 2, 2, 5, 2, 11, 31, 2, 17, 15073, 2, 47, 5, 5, 2, 2, 47, 2, 59, 2, 11, 5, 2, 2, 2, 5, 2, 2, 47, 2, 23, 2, 97, 349, 103, 2, 2, 67, 149, 2, 67

Offset: 1

Jianing Song, Nov 08 2019

nonn

In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not, where Pi(k,b)(n) is the number of primes <= n that are congruent to b modulo k. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Jun 26 2025]

This sequence gives the smallest primes q to violate the inequality Sum_{primes r <= q} Kronecker(r,p) <= 0, for p = prime(n).

			For prime(6) = 13, q = 2083 is the first case such that Sum_{primes r <= q} Kronecker(r,13) = 1 > 0, so a(6) = 2083.

Eric Weisstein's World of Mathematics, Kronecker Symbol
Wikipedia, Chebyshev's bias

Cf. A007350, A306499, A306500, A329225 (indices of these primes).

a(n) = if(n==2, 608981813029, if(n==3, 2082927221, my(p=prime(n), i=0); forprime(q=2, oo, i+=kronecker(q, p); if(i>0, return(q)))))

A329242 a(n) = Pi(8,3)(n) + Pi(8,5)(n) + Pi(8,7)(n) - 3Pi(8,1)(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, 0, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10

Offset: 1

Jianing Song, Nov 08 2019

sign

In general, assuming the strong form of the Riemann Hypothesis, if 0 < a, b < k are integers, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod k, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 19 2023]

Define the "Chebyshev's bias sequence mod k" to be sequence q(n), where q(n) = Sum_{b is a quadratic nonresidue mod k, gcd(b, k) = 1} Pi(k,b)(n) - (r-1)*(Sum_{a is a quadratic residue mod k, gcd(a, k) = 1} Pi(k,a)(n)), r is the number of solutions to x^2 == 1 (mod n), then this sequence is the "Chebyshev's bias sequence mod 8". Also the initial terms are nonnegative integers, a(n) is negative for some n ~ 10^28.127. See page 21 of the paper in Journal of Number Theory in the Links section below.

			Below 2000000, there are 37116 primes congruent to 1 mod 8, 37261 primes congruent to 3 mod 8, 37300 primes congruent to 5 mod 8 and 37255 primes congruent to 7 mod 8, so a(2000000) = 37261 + 37300 + 37255 - 3*37116 = 468.

Jianing Song, Table of n, a(n) for n = 1..10000
Bays, Carter & Ford, Kevin & Hudson, Richard & Rubinstein, Michael, Zeros of Dirichlet L-functions near the Real Axis and Chebyshev's Bias. Journal of Number Theory, Vol. 87, Issue 1, Mar 2001, pp. 54-76.
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias

Cf. A007350, A329243.

"Chebyshev's bias sequence mod k": A112632 (k=3), A038698 (k=4), A321862 (k=5), A321864 (k=7), A321865 (k=11).

a(n) = my(k=0); for(p=1, n, if(isprime(p)&&p>2, if(p%8==1, k-=3, k++))); k

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

Original entry on oeis.org

2, 3, 11, 71, 83, 223, 227, 503, 751, 1531, 1543, 1571, 1579, 1583, 4127, 5147, 5171, 5179, 5651, 6211, 11083, 11087, 11471, 11483, 11519, 11527, 11579, 11587, 17239, 17551, 17903, 17971, 35963, 36011, 39703, 39727

Offset: 1

Gerhard Kirchner, Nov 20 2021

nonn

The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see link "Oscillations of d(x)". If d(x) has no upper limit, the current sequence is infinite. Regarding the lower limit, see A349519.

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n  x  pi(x,4,3) pi(x,4,1)  d(x)=n-1?
  - --  --------- ---------  ---------
  1  2       0         0     0=0  true   a(1)  = 2
  2  3       1         0     1=1  true   a(2)  = 3
  3  5       1         1     0=2 false   a(3) != 5
  ...........................
  3 11       3         1     2=2  true   a(3)  = 11

Gerhard Kirchner, Oscillations of d(x)
Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A038691, A007350.

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 40,
  /* returns nmax terms */
while nsum then(n:n+1, sum:su, w: append(w,[p]) ) ) ,
w);

A381902 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1), while the total number of prime factors, counted with multiplicity, of the form 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 5, 15, 3, 12, 16, 20, 14, 26, 13, 39, 9, 30, 25, 35, 7, 28, 32, 34, 17, 51, 18, 40, 22, 50, 24, 38, 52, 44, 55, 60, 58, 29, 87, 21, 70, 64, 68, 46, 74, 37, 111, 33, 75, 45, 65, 78, 36, 80, 48, 82, 41, 123, 42, 91, 104, 56, 100, 62, 31

Offset: 1

Scott R. Shannon, Mar 09 2025

Unlike the EKG sequence A064413 the prime terms are not in their natural order, and the terms preceding and following such terms can be large multiples of the prime. The terms overall are distributed over multiple lines, with the primes falling on at least two lines; see the attached colored image. Due to the term selection rules numbers which have a sum of prime factor exponents for prime factors of the form 4*k+1 and 4*k+3 which differ by 3 or more can never appear, the smallest such number being 27.
In the first 100000 terms the fixed points are 1, 2, 88, 118, 304, 786, 826. It is likely no more exist.
There are five dominant lines on the graph of the first 100000 terms. They can be characterized as follows, from the highest sloped L1 to the lowest sloped L5, considering terms within 1% of the fitted equations. The approximate slopes of the five lines are 2.1284, 1.476, 1.4190, 1.06845, and 0.70947, so that the normalized slopes of L1, L3, L4 and L5 are 3, 2, 3/2 and 1. L5 has essentially has only prime terms, while the others essentially have none. The 5 lines encompass approx. 97% of terms in the range 50K-100K. - Bill McEachen, Aug 21 2025

			a(5) = 8 as the total number of prime factors of the form 4*k+1 and 4*k+3 for the first four terms is 0 and 1 respectively, thus a(5) cannot contain a single prime factor of the form 4*k+3. This eliminates 3 as a candidate, leaving 8 as the smallest available number that has no such prime factors and shares a factor with a(4) = 6. This is the first term to differ from A064413.
a(7) = 5 as the total number of prime factors of the form 4*k+1 and 4*k+3 for the first six terms is 1 and 1 respectively, thus a term can be chosen that contains a single odd prime factor, and 5 is the smallest unused term that shares a factor with a(6) = 10.

Scott R. Shannon, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Image of the first 100000 terms. The colors are graduated across the spectrum to show the total number of prime factors of each term, with red being one prime factor. The thin green line is a(n) = n.
Wikipedia, Chebyshev's bias.

Cf. A064413, A382091, A007350, A038698, A027748.

A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

Original entry on oeis.org

0, 1, 1, 2, 3, 2, 2, 3, 3, 5, 6, 5, 6, 2, 7, 5, 6, 8, 7, 7, 7, 12, 10, 10, 11, 11, 12, 10, 10, 12, 11, 13, 10, 10, 10, 10, 9, 8, 7, 9, 3, 4, 4, 4, 11, 13, 15, 17, 19, 19, 22, 19, 16, 13, 17, 16, 15, 16, 14, 17, 16, 21, 24, 19, 19, 13, 17, 17, 19, 19, 16, 11, 13, 13, 22, 19, 19, 17, 22

Offset: 1

Clark Kimberling and Robert G. Wilson v, May 04 2006

sign

Zero occurs infinitely often as do the negative numbers.
First occurrence of a(n) beginning with 0: 1, 2, 4, 5, 42, 10, 11, 15, 18, 37, 23, 25, 22, 32, 59, 47, 53, 48, 83, 49, 110, 62, 51, 82, 63, 170, ...,
The first negative term is at n=1473. - T. D. Noe, Apr 09 2009

T. D. Noe, Table of n, a(n) for n=1..2000

Cf. A022757, A007350, A007351, A038691.

(Select[1 + 4Range@245, PrimeQ@# &] - Select[ -1 + 4Range@225, PrimeQ@# &] - 2)/4

a(n) = (A002144(n) - A002145(n) - 2)/4.

A267135 a(n) = n minus the number of primes of form 4m + 1 that are less than n-th prime of form 4m + 3.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 2, 2, 3, 3, 4, 2, 3, 2, 3, 3, 3, 3, 4, 4, 4, 3, 4, 5, 6, 5, 5, 5, 5, 4, 4, 5, 4, 4, 3, 4, 4, 5, 2, 2, 2, 3, 1, 2, 3, 4, 5, 6, 7, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 7, 7, 5, 6, 5, 6, 6, 7, 8, 5, 4, 4, 5, 5, 4, 5, 4, 5, 6, 7, 8, 6, 6, 7, 7, 8, 6, 6, 6, 6, 6, 6, 7, 6, 5, 5, 5, 6, 6

Offset: 1

Gionata Neri, Jan 10 2016

sign

a(25191) = -3 is the first negative term. - Robert Israel, Jan 12 2016

Robert Israel, Table of n, a(n) for n = 1..10000
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.

Cf. A002144, A002145, A007350.

N:= 10000: # to use primes up to N P1:= select(isprime, [seq(i,i=1..N,4)]):
P3:= select(isprime, [seq(i,i=3..N,4)]):
V:= Vector(N):
for n from 2 to nops(P1) do
  V[P1[n-1]..P1[n]-1]:=n-1
od:
V[P1[nops(P1)]..N]:= nops(P1);
seq(n - V[P3[n]],n=1..nops(P3)); # Robert Israel, Jan 11 2016

nn = 10000;
P1 = Select[Range[1, nn, 4], PrimeQ];
P3 = Select[Range[3, nn, 4], PrimeQ];
V = Table[0, nn];
For[n = 2, n <= Length[P1], n++,
  V[[P1[[n-1]] ;; P1[[n]]-1]] = n-1
];
V[[P1[[Length[P1]]] ;; nn]] = Length[P1];
Table[n - V[[P3[[n]]]], {n, 1, Length[P3]}] (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)

A349519 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = 1-n where pi(x,4,k) is the number of primes 4*j + k <= x.

Original entry on oeis.org

2, 26861, 616897, 616909, 616933, 623641, 623653, 623669, 623681, 12315529, 12315581, 12315613, 12315617, 12362653, 12362657, 12362717, 12362741, 12362981, 12362989, 12365033, 12365057, 12365153, 12365173, 12365201, 12366589, 951821281

Offset: 1

Gerhard Kirchner, Nov 20 2021

nonn

The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see A349518, "Oscillations of d(x)". If d(x) has no lower limit, the current sequence is infinite. Regarding the upper limit, see A349518.

Note the gaps between 2, 26861 and 616897, 623681 and 12315529, 12366589 and 951821281.

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n     x  pi(x,4,3) pi(x,4,1)   d(x)=1-n?
  - -----  --------- ---------  -----------
  1     2       0         0      0=0  true   a(1)  = 2
  2     3       1         0      1=-1 false  a(2) != 3
  2     5       1         1      2=-1 false  a(2) != 5
  ...........................
  2 26861    1472      1473     -1=-1 true   a(3)  = 26861

Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Cf. A038691, A007350.

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 25,
  /* returns nmax terms */
while n

cp's OEIS Frontend

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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A329224 a(n) is the smallest prime q such that Sum_{primes r <= q} Kronecker(r,prime(n)) > 0 (or equivalently, Sum_{primes r <= q} Kronecker(r,prime(n)) = 1), or 0 if no such prime exists.

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PARI

A329242 a(n) = Pi(8,3)(n) + Pi(8,5)(n) + Pi(8,7)(n) - 3*Pi(8,1)(n), where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

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PARI

A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

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Maxima

A381902 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1), while the total number of prime factors, counted with multiplicity, of the form 4*k+1 and 4*k+3 for all terms a(1)..a(n) never differs by more than 1.

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A118480 (n-th 4k+1 prime minus n-th 4k+3 prime minus two)/4.

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Mathematica

Formula

A267135 a(n) = n minus the number of primes of form 4m + 1 that are less than n-th prime of form 4m + 3.

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Maple

Mathematica

A329242 a(n) = Pi(8,3)(n) + Pi(8,5)(n) + Pi(8,7)(n) - 3Pi(8,1)(n), where Pi(a,b)(x) denotes the number of primes in the arithmetic progression ak + b less than or equal to x.

A381902 a(1) = 1, a(2) = 2; for n > 2, a(n) is the smallest unused positive number such that a(n) shares a factor with a(n-1), while the total number of prime factors, counted with multiplicity, of the form 4k+1 and 4k+3 for all terms a(1)..a(n) never differs by more than 1.