The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2. 22
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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] 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. 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". - Jianing Song, Nov 24 2018 LINKS 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 FORMULA 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 EXAMPLE 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). MATHEMATICA a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a = a = 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]] (* Harvey P. Dale, Jun 06 2013 *) PROG (Haskell) a112632 n = a112632_list !! (n-1) a112632_list = scanl1 (+) \$ map negate a134323_list -- Reinhard Zumkeller, Sep 16 2014 (PARI) a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ Jianing Song, Nov 24 2018 CROSSREFS 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). Sequence in context: A023134 A304536 A272863 * A254575 A355402 A275344 Adjacent sequences:  A112629 A112630 A112631 * A112633 A112634 A112635 KEYWORD sign,nice AUTHOR Roger Hui, Dec 22 2005 STATUS approved

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

Last modified September 26 20:57 EDT 2022. Contains 357050 sequences. (Running on oeis4.)