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

Author

Martin Renner, May 28 2012

Keywords

• nonn

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Liouville Function

Crossrefs

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

Programs

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 *)

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

Extensions

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