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.

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

Offset

1,1

Comments

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.

References

Adolf J. Hildebrand, Multiplicative properties of consecutive integers; pp. 103-118 in Analytic number theory, ed. by Y. Motohashi.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Y. Buttkewitz and C. Elsholtz, Patterns and complexity of multiplicative functions, Journal of the London Mathematical Society 84:3 (2011), pp. 578-594.

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

Subsequence of A221281. Cf. A008836, A221280.

Sequence in context: A140150 A166658 A268330 * A033702 A000797 A171168

Adjacent sequences: A221279 A221280 A221281 * A221283 A221284 A221285

Keyword

nonn

Author

Charles R Greathouse IV, Jan 09 2013

Status

approved