|
%I #18 May 13 2013 01:54:23
%S 17,27,28,29,33,41,42,54,55,70,84,85,93,107,132,133,140,141,142,143,
%T 158,162,170,171,172,173,179,190,201,202,203,213,214,215,216,217,218,
%U 241,242,243,247,268,269,270,279,280,281,282,283,294,295,296,310,326,327,339,366
%N Numbers n such that lambda(n) = lambda(n+1) = lambda(n+2) = lambda(n+3), where lambda(n) = A008836(n) is the Liouville function.
%C Hildebrand proved that each of the 8 possible 3-tuples of values +1 and -1 occur infinitely often as values of the Liouville function at consecutive arguments. It seems difficult to extend Hildebrand's result to patterns of length larger than 3. However, for results in this direction see Buttkewitz & Elsholtz.
%D Adolf J. Hildebrand, Multiplicative properties of consecutive integers; pp. 103-118 in Analytic number theory, ed. by Y. Motohashi.
%H Charles R Greathouse IV, <a href="/A221282/b221282.txt">Table of n, a(n) for n = 1..10000</a>
%H Y. Buttkewitz and C. Elsholtz, <a href="http://jlms.oxfordjournals.org/content/84/3/578">Patterns and complexity of multiplicative functions</a>, Journal of the London Mathematical Society 84:3 (2011), pp. 578-594.
%e a(1) = 17 because 17, 18, 19, 20 each have an odd number of prime factors (counted with repetition, 1, 3, 1, 3, respectively) and this is the first integer for which this is true.
%t Select[Range[400], Length[Union[LiouvilleLambda[Range[#, # + 3]]]] == 1 &] (* _Alonso del Arte_, Jan 09 2013 *)
%o (PARI) is(n)=my(k=(-1)^bigomega(n)); k==(-1)^bigomega(n+1) && k==(-1)^bigomega(n+2) && k==(-1)^bigomega(n+3)
%Y Subsequence of A221281. Cf. A008836, A221280.
%K nonn
%O 1,1
%A _Charles R Greathouse IV_, Jan 09 2013
|
