%I A086971
%S A086971 0,0,0,1,0,1,0,1,1,1,0,2,0,1,1,1,0,2,0,2,1,1,0,2,1,1,1,2,0,3,0,1,1,1,1,
%T A086971 3,0,1,1,2,0,3,0,2,2,1,0,2,1,2,1,2,0,2,1,2,1,1,0,4,0,1,2,1,1,3,0,2,1,3,
%U A086971 0,3,0,1,2,2,1,3,0,2,1,1,0,4,1,1,1,2,0,4,1,2,1,1,1,2,0,2,2,3,0,3
%N A086971 a(n) = the number of distinct semiprime divisors of n.
%D A086971 Bender, E. A. and Goldman, J. R., On the Applications of Moebius Inversion 
               in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
%D A086971 Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the 
               Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.
%H A086971 M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/
               0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 
               226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A086971 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A086971 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Semiprime.html">Semiprime</a>
%H A086971 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               DivisorFunction.html">Divisor Function</a>
%H A086971 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               MoebiusTransform.html">Moebius Transform</a>.
%F A086971 If n = p1^e1 * p2^e2 * ... * pj^ej for primes p1, p2, ..., pj and integer 
               exponents e1, e2, ..., ej, then a(n) = |{k: ek >=2}| + T(j-1) where 
               T(k) is the k-th triangular number A000217(k). The proof follows 
               from the observation that any prime factor is either the square of 
               a prime if that prime squared is a factor of n, or the product of 
               2 distinct primes in the factorization of n, which is the binomial 
               coefficient C(j, 2) = T(j-1). - Jonathan Vos Post (jvospost3(AT)gmail.com), 
               Dec 08 2004
%F A086971 a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               May 02 2005
%F A086971 a(n) = omega(n/core(n)) + binomial(omega(n),2) = A001221(n/A007913(n)) 
               + binomial(A001221(n),2) = A056170(n) + A079275(n). - Rick L. Shepherd 
               (rshepherd2(AT)hotmail.com), Mar 06 2006
%F A086971 This is the inverse Moebius transform of A064911. - Jonathan Vos Post 
               (jvospost3(AT)gmail.com), Dec 08 2004
%o A086971 (PARI) /* These definitions of a(n) are equivalent. */ a(n) = sumdiv(n,
               d,bigomega(d)==2) a(n) = f=factor(n); j=matsize(f)[1]; sum(m=1,j,
               f[m,2]>=2) + binomial(j,2) a(n) = f=factor(n); j=omega(n); sum(m=1,
               j,f[m,2]>=2) + binomial(j,2) a(n) = omega(n/core(n)) + binomial(omega(n),
               2) - Rick L. Shepherd (rshepherd2(AT)hotmail.com), Mar 06 2006
%Y A086971 Cf. A001358, A064911, A001221, A000005, A000010, A004018.
%Y A086971 Cf. A007913, A056170, A079275, A001222.
%Y A086971 Sequence in context: A063962 A084114 A110475 this_sequence A088434 A034178 
               A131341
%Y A086971 Adjacent sequences: A086968 A086969 A086970 this_sequence A086972 A086973 
               A086974
%K A086971 nonn
%O A086971 1,12
%A A086971 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 22 2003
%E A086971 Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 28 2006