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

Author

Hans Havermann

Keywords

• sign
• easy
• nice
• hear

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)
• Vladimir Pletser, Chart for n=1..100000

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.

List of primes p such that a(p) = 0 is A007351. List of primes p such that a(p) < 0 is A199547. List of primes p such that a(p) = -1 is A051025. List of integers k such that a(prime(k)) = -1 is A051024. - Ya-Ping Lu, Jan 18 2025

Programs

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 *)

PARI

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

Python

from sympy import nextprime; a, p = 0, 2; R = [a]
for _ in range(2,88): p=nextprime(p); a += p%4-2; R.append(a)
print(*R, sep = ', ')  # Ya-Ping Lu, Jan 18 2025

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)

a(n) = a(n-1) + prime(n) (mod 4) - 2, n >= 2. - Ya-Ping Lu, Jan 18 2025