A212818 Numbers up to 10^n with an even number of not necessarily distinct prime factors, or positive Liouville function.

1, 5, 49, 493, 4953, 49856, 499735, 4999579, 49998058, 499987392, 4999941987, 49999828888, 499999738687, 4999999516711

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Eric Weisstein's World of Mathematics, Liouville Function

Formula

a(n) = A011557(n) - A212819(n).

a(n) = (10^n)/2 + A090410(n)/2. - Donovan Johnson, May 30 2012

a(n) = A055037(10^n). - Ray Chandler, May 30 2012

Example

a(1) = 5 since up to 10 there are the five numbers 1, 4, 6, 9, 10 with an even number of prime factors, or positive Liouville function.

Maple

zg:=0: zu:=0: G:=[]: U:=[]: k:=0:
for i from 1 to 10^8 do if numtheory[bigomega](i) mod 2 = 0 then zg:=zg+1: else zu:=zu+1: fi: if i=10^k then G:=[op(G), zg]: U:=[op(U), zu]: k:=k+1: fi: od:
print(G);

Mathematica

Table[Length[Select[Range[10^n], EvenQ[PrimeOmega[#]] &]], {n, 0, 5}] (* Alonso del Arte, May 28 2012 *)
Table[Count[LiouvilleLambda[Range[10^n]], 1], {n, 0, 5}] (* Ray Chandler, May 30 2012 *)

Crossrefs

Cf. A055037 (goes up to n rather than 10^n), A002819, A008836, A028260, A065043, A090410.

Sequence in context: A096596 A001079 A146311 * A195206 A081474 A370097

Adjacent sequences: A212815 A212816 A212817 * A212819 A212820 A212821

Keyword

nonn

Author

Martin Renner, May 28 2012

Extensions

a(9)-a(13) from Donovan Johnson, May 30 2012

Status

approved