A321856 -id:A321856 - cp's OEIS frontend

A066520 Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n.

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

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jan 05 2002

Although the initial terms are nonnegative, it has been proved that infinitely many terms are negative. The first two are a(26861)=a(26862)=-1. Next there are 3404 values of n in the range 616841 to 633798 with a(n)<0. Then 27218 values in the range 12306137 to 12382326.

Partial sums of A151763. - Reinhard Zumkeller, Feb 06 2014

T. D. Noe, Table of n, a(n) for n=1..30000 (enough terms to show the first dip into negative territory)
Carter Bays and Richard H. Hudson, Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes, Mathematics of Computation, 69 (2000) 861-866.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.

Cf. A066339, A066490, A007350, A051024, A051025.
Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), this sequence (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a066520 n = a066520_list !! (n-1)
a066520_list = scanl1 (+) $ map (negate . a151763) [1..]
-- Reinhard Zumkeller, Feb 06 2014

a[n_] := Length[Select[Range[3, n, 4], PrimeQ]]-Length[Select[Range[1, n, 4], PrimeQ]]
f[n_]:=Module[{c=Mod[n,4]},Which[!PrimeQ[n],0,c==3,1,c==1,-1]]; Join[{0,0}, Accumulate[Array[f,110,3]]] (* Harvey P. Dale, Mar 03 2013 *)

a(n) = A066490(n) - A066339(n).

a(2*n+1) = a(2*n+2) = -A156749(n). - Jonathan Sondow, May 17 2013

Edited by Dean Hickerson, Mar 05 2002

A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

Original entry on oeis.org

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

Offset: 1

Roger Hui, Dec 22 2005

Cumulative sums of A134323, negated. The first negative term is a(23338590792) = -1 for the prime 608981813029. See page 4 of the paper by Granville and Martin. - T. D. Noe, Jan 23 2008 [Corrected by Jianing Song, Nov 24 2018]

See the comment about "Chebyshev's bias" in A321856. - Jianing Song, Nov 24 2018

			a(1) = 1 because 2 == -1 (mod 3).
a(2) = 1 because 3 == 0 (mod 3) and does not change the counting.
a(3) = 2 because 5 == -1 (mod 3).
a(4) = 1 because 7 == 1 (mod 3).

T. D. Noe, Table of n, a(n) for n = 1..10000
A. Granville and G. Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), pp. 1-33.
Wikipedia, Chebyshev's bias

Cf. A007352, A098044, A102283, A134323.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), this sequence (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a112632 n = a112632_list !! (n-1)
a112632_list = scanl1 (+) $ map negate a134323_list
-- Reinhard Zumkeller, Sep 16 2014

a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}]  (* Jean-François Alcover, Jul 24 2012 *)
Accumulate[Which[IntegerQ[(#+1)/3],1,IntegerQ[(#-1)/3],-1,True,0]& /@ Prime[ Range[100]]] (* Harvey P. Dale, Jun 06 2013 *)

a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ Jianing Song, Nov 24 2018

a(n) = -Sum_{primes p<=n} Legendre(prime(i),3) = -Sum_{primes p<=n} Kronecker(-3,prime(i)) = -Sum_{i=1..n} A102283(prime(i)). - Jianing Song, Nov 24 2018

A071838 a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 08 2002

a(n) is the number of odd primes <= n that have 2 as a quadratic nonresidue minus the number of primes <= n that have 2 as a quadratic residue. See the comments about "Chebyshev's bias" in A321861. - Jianing Song, Nov 24 2018
Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1?
The first negative term occurs at a(11100143) = -1. - Jianing Song, Nov 08 2019

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Wikipedia, Chebyshev's bias

Cf. A091337.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), this sequence (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *)

for(n=1,200,print1(sum(i=1,n,if((i*isprime(i)-3)%8,0,1)+if((i*isprime(i)-5)%8,0,1)-if((i*isprime(i)-1)%8,0,1)-if((i*isprime(i)-7)%8,0,1)),", ")) \\ Program fixed by Jianing Song, Nov 08 2019

a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018

a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018

Edited by Peter Munn, Nov 19 2023

A320857 a(n) = Pi(8,5)(n) + Pi(8,7)(n) - Pi(8,1)(n) - Pi(8,3)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 24 2018

sign

a(n) is the number of odd primes <= n that have -2 as a quadratic nonresidue minus the number of primes <= n that have -2 as a quadratic residue.
It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 212 ones are negative.
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 18 2023]
Here, although 3 is not a quadratic residue modulo 8, for most n we have Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) - Pi(8,3)(n), Pi(8,3)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,5)(n) and Pi(8,5)(n) + Pi(8,7)(n) > Pi(8,1)(n) + Pi(8,7)(n).

			Pi(8,1)(200) = 8, Pi(8,5)(200) = 13, Pi(8,3)(200) = Pi(8,7)(200) = 12, so a(200) = 13 + 12 - 8 - 12 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias.

Cf. A007350, A188510.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), this sequence (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)

a(n) = -sum(i=1, n, isprime(i)*kronecker(-2, i))

from sympy import isprime; from numpy import sign
def A320857(n): return sum(isprime(i)*(i%2)*sign(i%8-4) for i in range(1,n+1)) # Ya-Ping Lu, Jan 25 2025

a(n) = -Sum_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).

A320858 a(n) = A320857(prime(n)).

Original entry on oeis.org

0, -1, 0, 1, 0, 1, 0, -1, 0, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 1, 0, 1, 2, 1, 2, 1, 2, 1, 0, -1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 4, 3, 2, 3, 2, 3, 4, 5, 4, 5, 6, 7, 6, 5, 4, 5, 6, 5, 6, 5, 4

Offset: 1

Jianing Song, Nov 24 2018

sign

Among the first 10000 terms there are only 100 negative ones. See the comments about "Chebyshev's bias" in A320857.

			prime(46) = 199, Pi(8,1)(199) = 8, Pi(8,5)(199) = 13, Pi(8,3)(199) = Pi(8,7)(199) = 12, so a(46) = 13 + 12 - 8 - 12 = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Chebyshev's bias.

Cf. A188510.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), this sequence (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a[n_] := -Sum[KroneckerSymbol[-2, Prime[i]], {i, 1, n}];
Array[a, 100] (* Jean-François Alcover, Dec 28 2018, from PARI *)

a(n) = -sum(i=1, n, kronecker(-2, prime(i)))

a(n) = -Sum_{i=1..n} Kronecker(prime(i),2) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A091337(prime(i)).

A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(n) where Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of primes <= n that are quadratic nonresidues modulo 5 minus the number of primes <= n that are quadratic residues modulo 5.
a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(2082927221) = -1. - Jianing Song, Nov 08 2019
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			Pi(5,1)(100) = Pi(5,4)(100) = 5, Pi(5,2)(100) = Pi(5,3)(100) = 7, so a(100) = 7 + 7 - 5 - 5 = 4.

Wikipedia, Chebyshev's bias

Cf. A080891.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), this sequence (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, isprime(i)*kronecker(5, i))

a(n) = -Sum_{primes p<=n} Legendre(p,5) = -Sum_{primes p<=n} Kronecker(5,p) = -Sum_{primes p<=n} A080891(p).

Edited by Peter Munn, Nov 18 2023

A321859 Number of primes congruent to 3, 5, 6 modulo 7 and <= n minus number of primes congruent to 1, 2, 4 modulo 7 and <= n.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of primes <= n that are quadratic nonresidues modulo 7 minus the number of primes <= n that are quadratic residues modulo 7.
The first 10000 terms (except for a(2)) are nonnegative. a(p) = 0 for primes p = 3, 11, 211, 3371, 3389, ... The earliest negative term (besides a(2)) is a(48673) = -1. Conjecturally infinitely many terms should be negative.
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			Below 100, there are 10 primes congruent to 1, 2, 4 modulo 7 and 14 primes congruent to 3, 5, 6 modulo 7, so a(100) = 14 - 10 = 4.

Wikipedia, Chebyshev's bias

Cf. A175629.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), this sequence (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

Accumulate[Table[Which[PrimeQ[n]&&MemberQ[{3,5,6},Mod[n,7]],1,PrimeQ[ n] && MemberQ[ {1,2,4},Mod[ n,7]],-1,True,0],{n,90}]] (* Harvey P. Dale, Apr 28 2022 *)

a(n) = -sum(i=1, n, isprime(i)*kronecker(-7, i))

a(n) = -Sum_{primes p<=n} Legendre(p,7) = -Sum_{primes p<=n} Kronecker(-7,p) = -Sum_{primes p<=n} A175629(p).

A321860 Number of primes congruent to 2, 6, 7, 8, 10 modulo 11 and <= n minus number of primes congruent to 1, 3, 4, 5, 9 modulo 11 and <= n.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of primes <= n that are quadratic nonresidues modulo 11 minus the number of primes <= n that are quadratic residues modulo 11.
It seems that there are more negative terms here than in some other sequences mentioned in crossrefs; nevertheless, among the first 10000 terms, only 138 ones are negative.
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			Below 200, there are 20 primes congruent to 1, 3, 4, 5, 9 modulo 11 and 23 primes congruent to 2, 6, 7, 8, 10 modulo 11, so a(200) = 23 - 20 = 3.

Wikipedia, Chebyshev's bias

Cf. A112632.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: this sequence (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, isprime(i)*kronecker(-11, i))

a(n) = -Sum_{primes p<=n} Legendre(p,11) = -Sum_{primes p<=n} Kronecker(-11,p) = -Sum_{primes p<=n} A011582(p).

A321862 a(n) = A321857(prime(n)).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 3, 4, 5, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 6, 5, 4, 5, 6, 5, 6, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 5, 4

Offset: 1

Jianing Song, Nov 20 2018

sign

The first 10000 terms are positive, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(102091236) = -1. - Jianing Song, Nov 08 2019
Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			prime(25) = 97, Pi(5,1)(97) = Pi(5,4)(97) = 5, Pi(5,2)(97) = Pi(5,3)(97) = 7, so a(25) = 7 + 7 - 5 - 5 = 4.

Wikipedia, Chebyshev's bias

Cf. A080891.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), A321864 (d=-7), A038698 (d=-4), A112632 (d=-3), this sequence (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(5, prime(i)))

a(n) = -Sum_{i=1..n} Legendre(prime(i),5) = -Sum_{primes p<=n} Kronecker(2,prime(i)) = -Sum_{i=1..n} A080891(prime(i)).

Edited by Peter Munn, Nov 19 2023

A321864 a(n) = A321859(prime(n)).

Original entry on oeis.org

-1, 0, 1, 1, 0, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 0, 1, 0, 1, 2, 3, 2, 1, 2, 3, 4, 3, 4, 5, 4, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 5, 6, 5, 6, 5, 4, 5, 4, 5, 4, 5, 6, 5, 4, 5, 6, 5, 4

Offset: 1

Jianing Song, Nov 20 2018

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Among the first 10000 terms there are only 13 negative ones, with the earliest one (besides a(1)) being a(5006) = -1.

Please see the comment in A321856 describing "Chebyshev's bias" in the general case.

			prime(25) = 97. Among the primes <= 97, there are 10 ones congruent to 1, 2, 4 modulo 7 and 14 ones congruent to 3, 5, 6 modulo 7, so a(25) = 14 - 10 = 4.

Wikipedia, Chebyshev's bias

Cf. A175629.
Let d be a fundamental discriminant.
Sequences of the form "a(n) = -Sum_{primes p<=n} Kronecker(d,p)" with |d| <= 12: A321860 (d=-11), A320857 (d=-8), A321859 (d=-7), A066520 (d=-4), A321856 (d=-3), A321857 (d=5), A071838 (d=8), A321858 (d=12).
Sequences of the form "a(n) = -Sum_{i=1..n} Kronecker(d,prime(i))" with |d| <= 12: A321865 (d=-11), A320858 (d=-8), this sequence (d=-7), A038698 (d=-4), A112632 (d=-3), A321862 (d=5), A321861 (d=8), A321863 (d=12).

a(n) = -sum(i=1, n, kronecker(-7, prime(i)))

a(n) = -Sum_{primes p<=n} Legendre(prime(i),7) = -Sum_{primes p<=n} Kronecker(-7,prime(i)) = -Sum_{i=1..n} A175629(prime(i)).

cp's OEIS Frontend

A066520 Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n.

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A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

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A071838 a(n) = Pi(8,3)(n) + Pi(8,5)(n) - Pi(8,1)(n) - Pi(8,7)(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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A320857 a(n) = Pi(8,5)(n) + Pi(8,7)(n) - Pi(8,1)(n) - Pi(8,3)(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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A320858 a(n) = A320857(prime(n)).

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A321857 a(n) = Pi(5,2)(n) + Pi(5,3)(n) - Pi(5,1)(n) - Pi(5,4)(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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A321859 Number of primes congruent to 3, 5, 6 modulo 7 and <= n minus number of primes congruent to 1, 2, 4 modulo 7 and <= n.

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