cp's OEIS Frontend

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

%I A056912 #23 Jul 02 2025 16:02:00
%S A056912 3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,
%T A056912 101,103,105,107,109,113,127,131,137,139,149,151,157,163,165,167,173,
%U A056912 179,181,191,193,195,197,199,211,223,227,229,231,233,239,241,251,255
%N A056912 Odd squarefree numbers for which the number of prime divisors is odd.
%C A056912 Liouville function lambda(n) (A008836) is negative.
%C A056912 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
%C A056912 The asymptotic density of this sequence is 2/Pi^2 (A185197). - _Amiram Eldar_, Oct 06 2020
%D A056912 H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.
%H A056912 Charles R Greathouse IV, <a href="/A056912/b056912.txt">Table of n, a(n) for n = 1..10000</a>
%H A056912 H. Gupta, <a href="/A002556/a002556.pdf"> A formula for L(n)</a>, J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]
%e A056912 a(27) = 3*5*7 = 105 is the least nonprime.
%t A056912 Select[Range[3, 300], SquareFreeQ[#] && LiouvilleLambda[#] == -1 &] (* _Jean-François Alcover_, Jul 30 2013 *)
%t A056912 Select[Range[1, 255, 2], MoebiusMu[#] == -1 &] (* _Amiram Eldar_, Oct 06 2020 *)
%o A056912 (PARI) isok(n) = (n%2) && issquarefree(n) && (omega(n)%2) \\ _Michel Marcus_, Jun 15 2013
%o A056912 (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
%Y A056912 Cf. A056911, A056913, A008836, A026424, A028260, A185197.
%K A056912 easy,nonn
%O A056912 1,1
%A A056912 _James Sellers_, Jul 07 2000