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A038698 Surfeit of 4k-1 primes over 4k+1 primes, beginning with prime 2. 9

%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 4k-1 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 4N-1, where x[n] = 1 when n is prime otherwise x[n] is zero. The complex Fourier components of the n-th harmonic equals the complex number X[N] = -1 + j[pi(4k+1) - pi(4k-1)], where pi(4k+1) and pi(4k-1) are the number of primes of the form 4k+1 and 4k-1 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:=ct-1; 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 3k-1 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_

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Last modified August 17 20:01 EDT 2018. Contains 313817 sequences. (Running on oeis4.)