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Search: id:A072000
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| 0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 26
(list; graph; listen)
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OFFSET
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1,6
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COMMENT
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Number of k <= n such that bigomega(k) = 2.
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REFERENCES
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A. Hildebrand, On the number of prime factors of an integer, in Ramanujan Revisited (Urbana-Champaign, Ill., 1987), pp. 167-185, Academic Press, Boston, MA, 1988.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).
G. Tenenbaum, Introduction \`a la th\'eorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..40882
Eric Weisstein's World of Mathematics, Semiprime
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FORMULA
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Let Pi(x) denote the number of primes <= x (cf. A000720). Then 2*a(n) = Sum_{ primes p <= n/2 } Pi(n/p) + Pi(sqrt(n)). [Landau, p. 211]
Let Pi(x) denote the number of primes <= x (cf. A000720). Then a(n) = Sum_{i=1..Pi(sqrt(n))} (Pi(n/prime(i)) - i + 1). - Robert G. Wilson v Feb 07 2006
a(n) = card{ x <= n : bigomega(x) = 2 }.
Asymptotically a(n) ~ n*loglog(n)/log(n). [Landau, p. 211]
Let A be a positive integer. Then card{ x <= n : bigomega(x) = A } ~ (n/log(n))*log(log(n))^(A-1)/(A-1)! [Landau, p. 211]
a(n) = A072613(n)+A056811(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2007
a(n) = SUM[i=1..n] A064911(i). - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 30 2007
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MAPLE
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A072000 := proc(n) local sp, t ; sp := 0 ; for t from 1 to n do if numtheory[bigomega](t) = 2 then sp := sp+1 ; fi ; od ; sp ; end: for n from 1 to 180 do printf("%d, ", A072000(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 10 2007
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MATHEMATICA
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SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] -i + 1, {i, PrimePi@Sqrt@n}]; Array[SemiPrimePi, 78] (from Robert G. Wilson v (rgwv(at)rgwv.com), Jan 03 2006)
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PROGRAM
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(PARI) for(n=1, 100, print1(sum(i=1, n, if(bigomega(i)-2, 0, 1)), ", "))
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CROSSREFS
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Cf. A000720, A001358, A066265, A064911.
Sequence in context: A029131 A162351 A087816 this_sequence A157477 A006949 A055748
Adjacent sequences: A071997 A071998 A071999 this_sequence A072001 A072002 A072003
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KEYWORD
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easy,nonn
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AUTHOR
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Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 19 2002
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EXTENSIONS
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Edited by Robert G. Wilson v, Feb 15 2006
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