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

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

Offset: 1

Author

Roger Hui, Dec 22 2005

Keywords

• sign
• nice

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]

See the comment about "Chebyshev's bias" in A321856. - Jianing Song, Nov 24 2018

Examples

			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).
		

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

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).

Programs

Haskell

a112632 n = a112632_list !! (n-1)
a112632_list = scanl1 (+) $ map negate a134323_list
-- Reinhard Zumkeller, Sep 16 2014

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-François 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 *)

PARI

a(n) = -sum(i=1, n, kronecker(-3, prime(i))) \\ Jianing Song, Nov 24 2018

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