A212818 - OEIS

%I #17 May 30 2012 16:42:40

%S 1,5,49,493,4953,49856,499735,4999579,49998058,499987392,4999941987,

%T 49999828888,499999738687,4999999516711

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LiouvilleFunction.html">Liouville Function</a>

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

%F a(n) = (10^n)/2 + A090410(n)/2. - _Donovan Johnson_, May 30 2012

%F a(n) = A055037(10^n). - _Ray Chandler_, May 30 2012

%e 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.

%p zg:=0: zu:=0: G:=[]: U:=[]: k:=0:

%p 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:

%p print(G);

%t Table[Length[Select[Range[10^n], EvenQ[PrimeOmega[#]] &]], {n, 0, 5}] (* _Alonso del Arte_, May 28 2012 *)

%t Table[Count[LiouvilleLambda[Range[10^n]], 1], {n, 0, 5}] (* _Ray Chandler_, May 30 2012 *)

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

%K nonn

%O 0,2

%A _Martin Renner_, May 28 2012

%E a(9)-a(13) from _Donovan Johnson_, May 30 2012