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A038698 Surfeit of 4k-1 primes over 4k+1 primes, beginning with prime 2. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

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) - 2n. (Assuming that x mod 4 is a positive number.) - Thomas Ordowski, Sep 21 2012

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, A066520.

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

Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)).

Sequence in context: A029359 A173389 A241062 * A087991 A144095 A076092

Adjacent sequences:  A038695 A038696 A038697 * A038699 A038700 A038701

KEYWORD

sign,easy,nice

AUTHOR

Hans Havermann

STATUS

approved

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Last modified November 20 21:20 EST 2014. Contains 249754 sequences.