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A086971
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a(n) = the number of distinct semiprime divisors of n.
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5
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,12
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COMMENTS
| Smallest n with exactly k (k=0..12) semiprime divisors are: 1, 4, 12, 30, 60, 180, 210, 420, 1260, 6300, 2310, 4620, 13860. - Zak Seidov, Mar 31 2011
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REFERENCES
| Bender, E. A. and Goldman, J. R., On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
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.
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LINKS
| M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Semiprime
Eric Weisstein's World of Mathematics, Divisor Function
Eric Weisstein's World of Mathematics, Moebius Transform.
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FORMULA
| 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
a(n) = A106404(n) + A106405(n). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), May 02 2005
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
Inverse Moebius transform of A064911. - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 08 2004
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MATHEMATICA
| semiQ[n_]:=2==Plus@@Last/@FactorInteger[n]; sp=Select[Range[4, 40000], semiQ]; Table[Length[Intersection[sp, Divisors[n]]], {n, 1, 40000}]; (* Zak Seidov, Mar 31 2011 *)
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PROG
| (PARI) /* The following 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 */
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CROSSREFS
| Cf. A001358, A064911, A001221, A000005, A000010, A004018, A007913, A056170, A079275, A001222.
Sequence in context: A063962 A084114 A110475 * A088434 A205745 A034178
Adjacent sequences: A086968 A086969 A086970 * A086972 A086973 A086974
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KEYWORD
| nonn
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AUTHOR
| Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 22 2003
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 28 2006
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