A056912 Odd squarefree numbers for which the number of prime divisors is odd.

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 231, 233, 239, 241, 251, 255

Offset: 1

Author

James Sellers, Jul 07 2000

Keywords

• easy
• nonn

Comments

Liouville function lambda(n) (A008836) is negative.

m is a term iff mu(m)^m < 0 (A080323(a(n))<0), where mu is the Moebius function (A008683). - Reinhard Zumkeller, Feb 14 2003

The asymptotic density of this sequence is 2/Pi^2 (A185197). - Amiram Eldar, Oct 06 2020

Examples

			a(27) = 3*5*7 = 105 is the least nonprime.
		

References

• H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]

Crossrefs

Cf. A056911, A056913, A008836, A026424, A028260, A185197.

Programs

Mathematica

Select[Range[3, 300], SquareFreeQ[#] && LiouvilleLambda[#] == -1 &] (* Jean-François Alcover, Jul 30 2013 *)
Select[Range[1, 255, 2], MoebiusMu[#] == -1 &] (* Amiram Eldar, Oct 06 2020 *)

PARI

isok(n) = (n%2) && issquarefree(n) && (omega(n)%2) \\ Michel Marcus, Jun 15 2013

PARI

is(n)=if(n%2, my(f=factor(n)[,2]);n>1 && vecmax(f)<2 && #f%2, 0) \\ Charles R Greathouse IV, Jun 15 2013