|
|
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.
|
|
15
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,31
|
|
COMMENTS
|
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).
|
|
LINKS
|
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
|
|
FORMULA
|
a(n) = -Sum_{primes p<=n} Kronecker(-2,p) = -Sum_{primes p<=n} A188510(p).
|
|
EXAMPLE
|
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.
|
|
MATHEMATICA
|
Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[-2, #], 0] &, 88] (* Michael De Vlieger, Nov 25 2018 *)
|
|
PROG
|
(PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(-2, i))
|
|
CROSSREFS
|
Let d be a fundamental discriminant.
|
|
KEYWORD
|
sign
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|