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A066520 Number of primes of the form 4m+3 <= n minus number of primes of the form 4m+1 <= n. 9
0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

Although the initial terms are nonnegative, it has been proved that infinitely many terms are negative. The first two are a(26861)=a(26862)=-1. Next, there are 3404 values of n in the range 616841 to 633798 with a(n)<0. Then 27218 values in the range 12306137 to 12382326.

Partial sums of A151763. - Reinhard Zumkeller, Feb 06 2014

REFERENCES

Carter Bays & Richard H. Hudson, Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes, Mathematics of Computation, 69 (2000) 861-866.

LINKS

T. D. Noe, Table of n, a(n) for n=1..30000 (enough terms to show the first dip into negative territory)

Carter Bays & Richard H. Hudson, Zeros of Dirichlet L-Functions and Irregularities in the Distribution of Primes

A. Granville and G. Martin, Prime number races

FORMULA

a(n) = A066490(n) - A066339(n)

a(2*n+1) = a(2*n+2) = -A156749(n) (Jonathan Sondow, May 17 2013)

MATHEMATICA

a[n_] := Length[Select[Range[3, n, 4], PrimeQ]]-Length[Select[Range[1, n, 4], PrimeQ]]

f[n_]:=Module[{c=Mod[n, 4]}, Which[!PrimeQ[n], 0, c==3, 1, c==1, -1]]; Join[{0, 0}, Accumulate[Array[f, 110, 3]]] (* Harvey P. Dale, Mar 03 2013 *)

PROG

(Haskell)

a066520 n = a066520_list !! (n-1)

a066520_list = scanl1 (+) $ map (negate . a151763) [1..]

-- Reinhard Zumkeller, Feb 06 2014

CROSSREFS

Cf. A066339, A066490, A007350, A051024, A051025.

Cf. A156749 Sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues, Mar 26 2009]

Sequence in context: A051031 A181059 A111915 * A088526 A054535 A054534

Adjacent sequences:  A066517 A066518 A066519 * A066521 A066522 A066523

KEYWORD

sign,easy,nice,look

AUTHOR

Sharon Sela (sharonsela(AT)hotmail.com), Jan 05 2002

EXTENSIONS

Edited by Dean Hickerson, Mar 05 2002

STATUS

approved

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Last modified November 23 20:23 EST 2017. Contains 295141 sequences.