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