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Search: id:A086971
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| A086971 |
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a(n) = the number of distinct semiprime divisors of n. |
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+0
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; listen)
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OFFSET
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1,12
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REFERENCES
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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
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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
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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
This is the inverse Moebius transform of A064911. - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 08 2004
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PROGRAM
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(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
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CROSSREFS
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Cf. A001358, A064911, A001221, A000005, A000010, A004018.
Cf. A007913, A056170, A079275, A001222.
Sequence in context: A063962 A084114 A110475 this_sequence A088434 A034178 A131341
Adjacent sequences: A086968 A086969 A086970 this_sequence A086972 A086973 A086974
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Sep 22 2003
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EXTENSIONS
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Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Mar 28 2006
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