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A112632 Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2. 18
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[1] = a[2] = 1; Table[a[n], {n, 1, 100}]  (* Jean-Fran├žois Alcover, Jul 24 2012 *)

Accumulate[Which[IntegerQ[(#+1)/3], 1, IntegerQ[(#-1)/3], -1, True, 0]& /@ Prime[ Range[100]]] (* 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 A275344 A206826

Adjacent sequences:  A112629 A112630 A112631 * A112633 A112634 A112635

KEYWORD

sign,nice

AUTHOR

Roger Hui, Dec 22 2005

STATUS

approved

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Last modified January 20 18:46 EST 2019. Contains 319335 sequences. (Running on oeis4.)