%I
%S 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,
%T 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,
%U 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
%N Surfeit of 4k1 primes over 4k+1 primes, beginning with prime 2.
%C a(n) < 0 for infinitely many values of n.  _Benoit Cloitre_, Jun 24 2002
%C First negative value is a(2946) = 1, which is for prime 26861.  _David W. Wilson_, Sep 27 2002
%C The elements of this sequence can be found in the Discrete Fourier Transform X[f] of length 4N on the prime number sequence x[n] from n=0 to 4N1, where x[n] = 1 when n is prime otherwise x[n] is zero. The complex Fourier components of the nth harmonic equals the complex number X[N] = 1 + j[pi(4k+1)  pi(4k1)], where pi(4k+1) and pi(4k1) are the number of primes of the form 4k+1 and 4k1 less than 4N respectively.  Paul Mackenzie (paul.mackenzie(AT)ozemail.com.au), Jul 09 2010
%D Stan Wagon, The Power of Visualization, Front Range Press, 1994, p. 2.
%H T. D. Noe and N. J. A. Sloane, <a href="/A038698/b038698.txt">Table of n, a(n) for n = 1..20000</a>, Jun 24 2016 [First 10000 terms from T. D. Noe]
%F a(n) = sum(k=2..n, (1)^((prime(k)+1)/2)).  _Benoit Cloitre_, Jun 24 2002
%F a(n) = sum(k=1..n, prime(k) mod 4)  2n. (Assuming that x mod 4 is a positive number.)  _Thomas Ordowski_, Sep 21 2012
%F From _Antti Karttunen_, Oct 01 2017: (Start)
%F a(n) = A267098(n)  A267097(n).
%F a(n) = A292378(A000040(n)).
%F (End)
%p ans:=[0]; ct:=0; for n from 2 to 2000 do
%p p:=ithprime(n); if (p mod 4) = 3 then ct:=ct+1; else ct:=ct1; fi;
%p ans:=[op(ans),ct]; od: ans; # _N. J. A. Sloane_, Jun 24 2016
%t FoldList[Plus, 0, Mod[Prime[Range[2,110]], 4]  2]
%t Join[{0},Accumulate[If[Mod[#,4]==3,1,1]&/@Prime[Range[2,110]]]] (* _Harvey P. Dale_, Apr 27 2013 *)
%o (PARI) for(n=2,100,print1(sum(i=2,n,(1)^((prime(i)+1)/2)),","))
%Y Cf. A007350, A007351, A038691, A066520.
%Y Cf. A112632 (race of 3k1 and 3k+1 primes), A216057, A269364.
%Y Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)), A199547, A267097, A267098, A267107, A292378.
%K sign,easy,nice,hear
%O 1,5
%A _Hans Havermann_
