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A166510
Nearest integer to 1 / Sum_{p prime, 2^n < p <= 2^(n+1)} (Kronecker(-1/p)/p).
2
2, -3, 18, -71, -29, 40, -57, -61, 785, 671, -354, 2984, 10869, -1374, -13678, -4224, -3217, 9010, -20224, -23384, 21468, -34445, -23448, 165023, -466254, -146824, 174689, -399880
OFFSET
0,1
COMMENTS
The convergence of the quasi-alternating sum S = Sum_{prime p >= 2} (Kronecker(-1/p)/p) (= A166509), where Kronecker(-1/p) = -1 if p=3 (mod 4), and +1 else (essentially the same as A070750), can be estimated by grouping together the terms for primes between consecutive powers of 2. These "partial sums" indeed go to zero, so we list here the nearest integer to their inverse. (Nearest integer means floor(x+0.5), see examples.) For more links and references, see A166509.
LINKS
J. P. Benney, Is this a convergent series and if so what is its sum?, in primenumbers group, Oct 26 2009.
Julien Benney, Mark Underwood, Andrew J. Walker, David Broadhurst, Is this a convergent series and if so what is its sum?, digest of 12 messages in primenumbers Yahoo group, Oct 26 - Oct 30, 2009. [Cached copy]
EXAMPLE
a(0) = 2 is the inverse of 1/2. (For the prime p=2, Kronecker(-1/2) = +1.)
a(1) = -3 is the inverse of -1/3. (For the prime p=3, Kronecker(-1/p)=-1, as for any prime p=3 (mod 4).)
a(2) = 18 = round(17.5), since 1/5-1/7 = 2/35. (For the prime p=5, Kronecker(-1/p)=+1, as for any prime p=1 (mod 4).
a(3) = -71 = round(-71.5), since -1/11+1/13 = -2/143. (Here we summed over primes between 2^3=8 and 2^4=16. The negative half-integer value is rounded towards zero.)
MATHEMATICA
kron[p_] := If[Mod[p, 4] == 3, -1, 1]; a[n_] := (p = 2^n; s = 0; While[(p = NextPrime[p]) < 2^(n + 1), s += kron[p]/p]; Floor[1/s + 1/2]); a[0] = 2; Table[an = a[n]; Print[an]; an, {n, 0, 27}] (* Jean-François Alcover, Nov 09 2011, after Pari *)
PROG
(PARI) A166510(n)={ my(p=2^n, s=.); n=2<<n; while(n>=p=nextprime(p+1), s+=kronecker(-1, p)/p); round(1/s)}
CROSSREFS
Cf. A166509 (value of the sum S), A086239 (= 0.5 - S).
Sequence in context: A216628 A048047 A114165 * A288492 A184719 A076016
KEYWORD
more,sign
AUTHOR
M. F. Hasler, Oct 26 2009
STATUS
approved