

A221282


Numbers n such that lambda(n) = lambda(n+1) = lambda(n+2) = lambda(n+3), where lambda(n) = A008836(n) is the Liouville function.


3



17, 27, 28, 29, 33, 41, 42, 54, 55, 70, 84, 85, 93, 107, 132, 133, 140, 141, 142, 143, 158, 162, 170, 171, 172, 173, 179, 190, 201, 202, 203, 213, 214, 215, 216, 217, 218, 241, 242, 243, 247, 268, 269, 270, 279, 280, 281, 282, 283, 294, 295, 296, 310, 326, 327, 339, 366
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OFFSET

1,1


COMMENTS

Hildebrand proved that each of the 8 possible 3tuples 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.


REFERENCES

Adolf J. Hildebrand, Multiplicative properties of consecutive integers; pp. 103118 in Analytic number theory, ed. by Y. Motohashi.


LINKS



EXAMPLE

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.


MATHEMATICA

Select[Range[400], Length[Union[LiouvilleLambda[Range[#, # + 3]]]] == 1 &] (* Alonso del Arte, Jan 09 2013 *)


PROG

(PARI) is(n)=my(k=(1)^bigomega(n)); k==(1)^bigomega(n+1) && k==(1)^bigomega(n+2) && k==(1)^bigomega(n+3)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



