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A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2. 18

%I

%S 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,

%T 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,

%U 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

%N Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.

%C 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]

%C 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

%H T. D. Noe, <a href="/A112632/b112632.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Granville and G. Martin, <a href="http://www.jstor.org/stable/27641834">Prime number races</a>, Amer. Math. Monthly, 113 (No. 1, 2006), pp. 1-33.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chebyshev%27s_bias">Chebyshev's bias</a>

%F 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

%e a(1) = 1 because 2 == -1 (mod 3).

%e a(2) = 1 because 3 == 0 (mod 3) and does not change the counting.

%e a(3) = 2 because 5 == -1 (mod 3).

%e a(4) = 1 because 7 == 1 (mod 3).

%t a[n_] := a[n] = a[n-1] + If[Mod[Prime[n], 6] == 1, -1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}] (* _Jean-Fran├žois Alcover_, Jul 24 2012 *)

%t Accumulate[Which[IntegerQ[(#+1)/3],1,IntegerQ[(#-1)/3],-1,True,0]& /@ Prime[ Range[100]]] (* _Harvey P. Dale_, Jun 06 2013 *)

%o (Haskell)

%o a112632 n = a112632_list !! (n-1)

%o a112632_list = scanl1 (+) $ map negate a134323_list

%o -- _Reinhard Zumkeller_, Sep 16 2014

%o (PARI) a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ _Jianing Song_, Nov 24 2018

%Y Cf. A007352, A098044, A102283, A134323.

%Y Let d be a fundamental discriminant.

%Y 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).

%Y 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).

%K sign,nice

%O 1,3

%A _Roger Hui_, Dec 22 2005

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Last modified February 21 01:15 EST 2019. Contains 320364 sequences. (Running on oeis4.)