A321856 -id:A321856 - cp's OEIS frontend

A321865 a(n) = A321860(prime(n)).

1, 0, -1, 0, 0, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 0, 1, 0, -1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 1, 2, 3, 2, 3, 4, 5, 4, 3, 4, 5, 4, 3, 2, 3, 4, 3, 4, 3, 4, 3, 4, 5, 6, 5, 4, 5, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 7, 6, 5, 4, 5, 4, 3, 4, 3, 4, 3, 2

Offset: 1

Jianing Song, Nov 20 2018

sign

Among the first 10000 terms there are only 32 negative ones.

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

			prime(46) = 199. Among the primes <= 199, there are 20 ones congruent to 1, 3, 4, 5, 9 modulo 11 and 23 ones congruent to 2, 6, 7, 8, 10 modulo 11, so a(46) = 23 - 20 = 3.

Wikipedia, Chebyshev's bias

Cf. A011582.
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: this sequence (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, kronecker(-11, prime(i)))

a(n) = -Sum_{primes p<=n} Legendre(prime(i),11) = -Sum_{primes p<=n} Kronecker(-11,prime(i)) = -Sum_{i=1..n} A011582(prime(i)).

A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(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, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is the number of odd primes <= n that have 3 as a quadratic nonresidue minus the number of primes <= n that have 3 as a quadratic residue.
The first 10000 terms are nonnegative. a(p) = 0 for primes p = 2, 3, 13, 433, 443, 457, 479, 491, 503, 3541, ... The earliest negative term is a(61463) = -1. Conjecturally infinitely many terms should be 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 19 2023]
Here, although 11 is not a quadratic residue modulo 12, for most n we have Pi(12,7)(n) + Pi(12,11)(n) > Pi(12,1)(n) - Pi(12,5)(n), Pi(12,5)(n) + Pi(12,11)(n) > Pi(12,1)(n) + Pi(12,7)(n) and Pi(12,5)(n) + Pi(12,7)(n) > Pi(12,1)(n) + Pi(12,11)(n).

			Pi(12,1)(100) = 5, Pi(12,5)(100) = Pi(12,7)(100) = Pi(12,11)(100) = 6, so a(100) = 6 + 6 - 5 - 6 = 1.

Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias

Cf. A007350, A110161.
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), this sequence (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(12, i))

a(n) = -Sum_{primes p<=n} Kronecker(12,p) = -Sum_{primes p<=n} A110161(p).

A321861 a(n) = A071838(prime(n)).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

a(n) is positive for 2 <= n <= 10000, but conjecturally infinitely many terms should be negative.
The first negative term occurs at a(732722) = -1. - Jianing Song, Nov 08 2019
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. Pi(a,b)(x) denotes the number of primes in the arithmetic progression a*k + b less than or equal to x. This phenomenon is called "Chebyshev's bias". (See Wikipedia link and especially the links in A007350.) [Edited by Peter Munn, Nov 19 2023].
Here, although 7 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).

			prime(25) = 97, Pi(8,1)(97) = 5, Pi(8,3)(97) = 7, Pi(8,5)(97) = Pi(8,7)(97) = 6, so a(25) = 7 + 6 - 5 - 6 = 2.

Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Wikipedia, Chebyshev's bias

Cf. A007350, 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), 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), this sequence (d=8), A321863 (d=12).

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)).

Edited by Peter Munn, Nov 19 2023

A321863 a(n) = A321858(prime(n)).

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 20 2018

sign

Among the first 10000 terms there are only 291 negative ones, with the earliest one being a(6181) = -1. See the comments about "Chebyshev's bias" in A321858.

			prime(25) = 97, Pi(12,1)(97) = 5, Pi(12,5)(97) = Pi(12,7)(97) = Pi(12,11)(97) = 6, so a(25) = 6 + 6 - 5 - 6 = 1.

Wikipedia, Chebyshev's bias

Cf. A110161.
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), A321862 (d=5), A321861 (d=8), this sequence (d=12).

a(n) = -sum(i=1, n, kronecker(12, prime(i)))

a(n) = -Sum_{primes p<=n} Kronecker(12,prime(i)) = -Sum_{i=1..n} A110161(prime(i)).

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

A341765 Consider gaps between successive odd primes from 3 up to prime(n+2). Let k1 be number of gaps congruent to 2 (mod 6) and let k2 be number of gaps congruent to 4 (mod 6). Then a(n) = k1 - k2.

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2

Offset: 1

Artur Jasinski, Feb 19 2021

nonn

Theorem A: for all n, a(n) belongs to the set: {1,2}, for proof see A342156.

The indices n for which numbers of 1's and 2's in this sequence are equal are 2, 4, 6, 10, 12, 20, 36, 46, 48 and no other up to n=10^6.

			a(1)=1 because prime(2+1)-prime(2)=5-3=2 then the gap 2 is congruent to 2 mod 6, then k1=1 and k2=0 so k1 - k2 = 1.

Cf. A000230, A001223, A028334, A321856, A341952, A342156, A039701.

k1 = 0; k2 = 0; cc = {}; Do[
gap = Prime[n + 1] - Prime[n];
If[Mod[gap/2, 3] == 1, k1 = k1 + 1,
  If[Mod[gap/2, 3] == 2, k2 = k2 + 1]]; AppendTo[cc, k1 - k2];
If[k1 - k2 == 1, , If[k1 - k2 == 2, , Print[{n, k1 - k2}]]], {n, 2,
  105}]; cc

a(n) = {my(vp = vector(n+1, k, prime(k+1)), dp = vector(#vp-1, k, (vp[k+1] - vp[k])/2)); my(s=0); for (k=1, #dp, if ((dp[k]%3)==1, s++); if ((dp[k]%3) == 2, s--)); s;} \\ Michel Marcus, Feb 27 2021

a(n) = 3 - A039701(n+2). - Andrey Zabolotskiy, Nov 04 2024

Name edited by Andrey Zabolotskiy, Nov 04 2024

A306891 Primes p for which pi_{3,2}(p) < pi_{3,1}(p), 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, 608981813191, 608981813261, 608981813269, 608981813273, 608981813311, 608981813347, 608981813357, 608981813449, 608981813459, 608981813683, 608981813701, 608981813707, 608981813711, 608981813717, 608981813719, 608981813777, 608981813779

Offset: 1

Jianing Song, Mar 16 2019

nonn

Primes p such that Sum_{primes q <= p} Kronecker(-3,q) > 0.

Indices of negative terms in A321856. See also the comment about Chebyshev's bias in A321856.

David A. Corneth, Table of n, a(n) for n = 1..10000
C. Bays, R. H. Hudson, Details of the First Region of Integers x with pi_{3,2}(x) < pi_{3,1}(x), Mathematics of Computation 32(142), 1978, pp. 571-576.
Eric Weisstein's World of Mathematics, Chebyshev Bias
Wikipedia, Chebyshev's bias

Cf. A096630, A112632, A297006, A321856.

Cf. also A199547, A096628.

my(i=0); forprime(p=608981813029, 608981820000, i+=kronecker(-3, p); if(i>0, print1(p, ", ")))

a(n) = prime(A096630(n)).

cp's OEIS Frontend

A321865 a(n) = A321860(prime(n)).

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A321858 a(n) = Pi(12,5)(n) + Pi(12,7)(n) - Pi(12,1)(n) - Pi(12,11)(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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A321861 a(n) = A071838(prime(n)).

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A321863 a(n) = A321858(prime(n)).

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A329243 a(n) = Pi(8,3)(prime(n)) + Pi(8,5)(prime(n)) + Pi(8,7)(prime(n)) - 3*Pi(8,1)(prime(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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A341765 Consider gaps between successive odd primes from 3 up to prime(n+2). Let k1 be number of gaps congruent to 2 (mod 6) and let k2 be number of gaps congruent to 4 (mod 6). Then a(n) = k1 - k2.

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Mathematica

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A306891 Primes p for which pi_{3,2}(p) < pi_{3,1}(p), where pi_{m,a}(x) is the number of primes <= x which are congruent to a (mod m).

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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.