This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 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. - Jianing Song, Nov 24 2018 Although the initial terms are nonnegative, infinitely many terms should be negative. For which n does a(n) = -1? In general, assuming the strong form of RH, if 0 < a, b < k, gcd(a, k) = gcd(b, k) = 1, a is a quadratic residue and b is a quadratic nonresidue mod n, then Pi(k,b)(n) > Pi(k,a)(n) occurs more often than not. This phenomenon is called "Chebyshev's bias". - Jianing Song, Nov 24 2018 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..10000 Wikipedia, Chebyshev's bias FORMULA a(n) = -Sum_{primes p<=n} Kronecker(2,p) = -Sum_{primes p<=n} A091337(p). - Jianing Song, Nov 20 2018 MATHEMATICA Accumulate@ Array[-If[PrimeQ@ #, KroneckerSymbol[2, #], 0] &, 105] (* Michael De Vlieger, Nov 25 2018 *) PROG (PARI) 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)), ", ")) (PARI) a(n) = -sum(i=1, n, isprime(i)*kronecker(2, i)) \\ Jianing Song, Nov 24 2018 CROSSREFS 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). Sequence in context: A238643 A140193 A073741 * A157896 A156072 A215788 Adjacent sequences:  A071835 A071836 A071837 * A071839 A071840 A071841 KEYWORD easy,nonn AUTHOR Benoit Cloitre, Jun 08 2002 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified October 19 21:28 EDT 2019. Contains 328244 sequences. (Running on oeis4.)