cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Jianing Song, Nov 20 2018

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

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

Formula

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)).

Extensions

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

Views

Author

Jianing Song, Nov 20 2018

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

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

Formula

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)).

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 20 2018

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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).

Programs

  • PARI
    a(n) = -sum(i=1, n, kronecker(-11, prime(i)))

Formula

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)).

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

Views

Author

Jianing Song, Nov 20 2018

Keywords

Comments

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).

Examples

			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.

Links

Crossrefs

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).

Programs

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

Formula

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)).

Extensions

Edited by Peter Munn, Nov 19 2023
Previous Showing 11-14 of 14 results.

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