

A112632


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


21



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
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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. 133.
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[n1] + If[Mod[Prime[n], 6] == 1, 1, 1]; a[1] = a[2] = 1; Table[a[n], {n, 1, 100}] (* JeanFrançois 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 !! (n1)
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



