This paper by Churchouse and Good is relevant: http://www.ams.org/journals/mcom/1968-22-104/S0025-5718-1968-0240062-8/S0025... Victor Sent from my iPhone On Feb 10, 2013, at 20:24, Dan Asimov <dasimov@earthlink.net> wrote:
Keith's February puzzle led me to wonder:
Let pf(n) denote the number of prime factors of n, counting multiplicity.
Now define Q: Z -> {1,-1} via
Q(n) := (-1)^pf(n), n > 1,
and Q(1) := 0.
QUESTION: As N -> oo, does the sequence
S_N := { s(k) = Q(N+k) | k in Z+ }
approach being indistinguishable from the outcomes of a sequence of independent Bernoulli trials -- i.e. of repeated flips of a fair coin (if H = 1, T = -1) ???
Such questions are always philosophically tricky, since the outcomes of independent random trials will have probability 0 of conforming exactly to any number-theoretic function like Q.
Maybe a good statistical test would be essentially the one used to define when a number is "normal", base 2, i.e.:
---------------------------------------------------------------------- REPHRASED QUESTION: For every L in Z+, does the distribution of the values in the multiset
S_(N,L) := {{ Q(N+1),...,Q(N+L) }}
approach the distribution of L independent Bernoulli trials as N -> oo, ??? ----------------------------------------------------------------------
--Dan
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