A007421 Liouville's function: parity of number of primes dividing n (with multiplicity). (Formerly M0067)

2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset

1,1

References

J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 279.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

H. Gupta, On a table of values of L(n), Proceedings of the Indian Academy of Sciences. Section A, 12 (1940), 407-409. [Annotated scanned copy]

R. S. Lehman, On Liouville's function, Math. Comp., 14 (1960), 311-320.

Formula

a(n) = ((-1)^bigomega(n)+3)/2, where bigomega(n) is the number of prime divisors of the integer n counted with multiplicity.

a(n) = A065043(n) + 1.

a(n) = 2 - A001222(n) mod 2. - Reinhard Zumkeller, Nov 10 2011

Mathematica

a[1] = 2; a[n_] := ((-1)^Total[FactorInteger[n][[All, 2]]] + 3)/2; (* or, from version 7 on : *) a[n_] := Boole[ EvenQ[ PrimeOmega[n]]] + 1; (* or *) a[n_] := (LiouvilleLambda[n] + 3)/2; a[1] = 2; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Apr 08 2013, updated Jan 27 2015 *)

Prog

(Haskell)
a007421 = (2 -) . (`mod` 2) . a001222 -- Reinhard Zumkeller, Nov 10 2011

Crossrefs

Cf. A008836, A065043.

Sequence in context: A245933 A268318 A309414 * A239228 A346080 A103921

Adjacent sequences: A007418 A007419 A007420 * A007422 A007423 A007424

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Dec 01 2001

Status

approved