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A101638
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a(n) = the number of distinct 4-almost primes A014613 which are factors of n.
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14
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,48
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COMMENTS
| This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes 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 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-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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EXAMPLE
| a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.
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CROSSREFS
| Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.
Sequence in context: A094428 A194024 A082786 * A070141 A088722 A122180
Adjacent sequences: A101635 A101636 A101637 * A101639 A101640 A101641
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KEYWORD
| easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 10 2004
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