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A101606
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a(n) = the number of distinct "3-almost primes" (A014612) which are divisors of n.
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16
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0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 3, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 0, 3, 0, 0, 0, 2, 0, 3, 0, 1, 0, 0, 0, 2, 0, 1, 1, 2, 0, 1, 0, 2, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,24
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COMMENTS
| This is the inverse Moebius transform of A101605. If n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=3}| + ((j-1)/2)*|{k: ek>=2}| + C(j,3). The first term is the number of distinct cubes of primes in the factors of n (the first way of finding a 3-almost prime). The second term is the number of distinct squares of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 3-almost prime). The third term is the number of distinct products of 3 distinct primes, which is the number of combinations of j primes taken 3 at a time, A000292(j), (the third way of finding a 3-almost prime).
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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.
Eric Weisstein's World of Mathematics, Almost Prime.
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) = a(n) = |{k: ek>=3}| + ((j-1)/2)*|{k: ek>=2}| + C(j, 3). where C(j, 3) is the binomial coefficient A000292(j).
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EXAMPLE
| a(60) = 3 because the factors of 60 include these three 3-almost primes: 12 = 2 * 2 * 3; 20 = 2 * 2 * 5; 30 = 2 * 3 * 5.
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MAPLE
| isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: A101606 := proc(n) a :=0 ; for d in numtheory[divisors](n) do if isA014612(d) then a := a+1 ; fi; od: a ; end: for n from 1 to 120 do printf("%d, ", A101606(n)) ; od: [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009]
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CROSSREFS
| Cf. A101605, A014612, A001358, A064911, A001221, A000005, A000010, A004018, A000292.
Sequence in context: A144625 A025442 A128582 * A125005 A122179 A125203
Adjacent sequences: A101603 A101604 A101605 * A101607 A101608 A101609
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 09 2004
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EXTENSIONS
| Replaced a(48) by 2 and a(76) by 1. R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009
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