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A112632
Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.
23
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
1,3
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]
See the comment about "Chebyshev's bias" in A321856. - Jianing Song, Nov 24 2018
LINKS
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-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 *)
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
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: A304536 A272863 A373352 * A254575 A355402 A275344
KEYWORD
sign,nice
AUTHOR
Roger Hui, Dec 22 2005
STATUS
approved