A038698 Excess of 4k-1 primes over 4k+1 primes, beginning with prime 2.

0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 2, 3, 4, 5, 6, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 3, 4, 3, 4, 5, 4, 3, 4, 3, 4, 3, 2, 3, 4, 3, 4, 5, 4, 3, 2, 1, 2, 1, 2, 1, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 6

Offset

1,5

Comments

a(n) < 0 for infinitely many values of n. - Benoit Cloitre, Jun 24 2002

First negative value is a(2946) = -1, which is for prime 26861. - David W. Wilson, Sep 27 2002

References

Stan Wagon, The Power of Visualization, Front Range Press, 1994, p. 2.

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)

Formula

a(n) = Sum_{k=2..n} (-1)^((prime(k)+1)/2). - Benoit Cloitre, Jun 24 2002

a(n) = (Sum_{k=1..n} prime(k) mod 4) - 2*n (assuming that x mod 4 > 0). - Thomas Ordowski, Sep 21 2012

From Antti Karttunen, Oct 01 2017: (Start)

a(n) = A267098(n) - A267097(n).

a(n) = A292378(A000040(n)).

(End)

From Ridouane Oudra, Nov 04 2024: (Start)

a(n) = Sum_{k=2..n} i^(prime(k)+1), where i is the imaginary unit.

a(n) = Sum_{k=2..n} sin(3*prime(k)*Pi/2).

a(n) = Sum_{k=2..n} A163805(prime(k)).

a(n) = Sum_{k=2..n} A212159(k). (End)

Maple

ans:=[0]; ct:=0; for n from 2 to 2000 do
p:=ithprime(n); if (p mod 4) = 3 then ct:=ct+1; else ct:=ct-1; fi;
ans:=[op(ans), ct]; od: ans; # N. J. A. Sloane, Jun 24 2016

Mathematica

FoldList[Plus, 0, Mod[Prime[Range[2, 110]], 4] - 2]
Join[{0}, Accumulate[If[Mod[#, 4]==3, 1, -1]&/@Prime[Range[2, 110]]]] (* Harvey P. Dale, Apr 27 2013 *)

Prog

(PARI) for(n=2, 100, print1(sum(i=2, n, (-1)^((prime(i)+1)/2)), ", "))

Crossrefs

Cf. A007350, A007351, A038691, A051024, A066520.

Cf. A112632 (race of 3k-1 and 3k+1 primes), A216057, A269364.

Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)), A199547, A267097, A267098, A267107, A292378.

Cf. A163805, A212159.

Sequence in context: A333471 A360711 A284620 * A333590 A263233 A300623

Adjacent sequences: A038695 A038696 A038697 * A038699 A038700 A038701

Keyword

sign,easy,nice,hear

Author

Hans Havermann

Status

approved