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A038698
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Surfeit of 4k-1 primes over 4k+1 primes, beginning with prime 2.
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4
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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; internal format)
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OFFSET
| 1,5
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COMMENTS
| a(n)<0 for infinitely many values of n - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 24 2002
First negative value is a(p(2946)) = a(26861) = -1. - David W. Wilson, Sep 27, 2002.
Contribution from Paul Mackenzie (paul.mackenzie(AT)ozemail.com.au), Jul 09 2010: (Start)
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 nth 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.
(End)
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REFERENCES
| Stan Wagon, The Power of Visualization, Front Range Press, 1994, p. 2.
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
| a(n) = sum(k=2, n, (-1)^((prime(n)+1)/2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 24 2002
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MATHEMATICA
| FoldList[Plus, 0, Mod[Prime[Range[2, 110]], 4]-2]
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PROG
| (PARI) for(n=2, 100, print1(sum(i=2, n, (-1)^((prime(i)+1)/2)), ", "))
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CROSSREFS
| Cf. A007350, A007351, A038691, A066520.
Cf. A112632 (race of 3k-1 and 3k+1 primes)
Cf. A156749 Another sequence showing Chebyshev bias in prime races (mod 4). [From Daniel Forgues (squid(AT)zensearch.com), Mar 26 2009]
Sequence in context: A029376 A029359 A173389 * A087991 A144095 A076092
Adjacent sequences: A038695 A038696 A038697 * A038699 A038700 A038701
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KEYWORD
| sign,easy,nice
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AUTHOR
| Hans Havermann (gladhobo(AT)teksavvy.com)
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EXTENSIONS
| Moved Mathematica program into Mathematica field from Formula field [Harvey P. Dale, June 25 2011]
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