

A038698


Surfeit of 4k1 primes over 4k+1 primes, beginning with prime 2.


8



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 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


REFERENCES

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


LINKS

T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 1..20000, Jun 24 2016 [First 10000 terms from T. D. Noe]


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


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:=ct1; 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 *)


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 3k1 and 3k+1 primes).
Cf. A156749 (sequence showing Chebyshev bias in prime races (mod 4)).
Sequence in context: A029359 A173389 A241062 * A263233 A087991 A144095
Adjacent sequences: A038695 A038696 A038697 * A038699 A038700 A038701


KEYWORD

sign,easy,nice,hear


AUTHOR

Hans Havermann


STATUS

approved



