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A072613
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Number of numbers of the form p*q (p, q distinct primes) less than or equal to n.
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5
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0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 4, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 22, 22, 22, 22
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OFFSET
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1,10
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COMMENTS
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There was an old comment here that said a(n) was equal to A070548(n) - 1, but this is false (e.g. at n=210). - N. J. A. Sloane, Sep 10 2008
Number of squarefree semiprimes not exceeding n. - Wesley Ivan Hurt, May 25 2015
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REFERENCES
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G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 1995.
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LINKS
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Eric Weisstein's World of Mathematics, Semiprime.
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FORMULA
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a(n) = Sum_{p<sqrt(n)} (Pi(x/p)-Pi(p)), where Pi(n) is the prime counting function, A000720, and the sum is over all primes less than sqrt(n). [N-E. Fahssi, Mar 05 2009]
Asymptotically a(n) ~ (n/log(n))log(log(n)) [G. Tenenbaum pp. 200--].
a(n) = Sum_{i<=n | Omega(i)=2} mu(i). - Wesley Ivan Hurt, Jan 05 2013, revised May 25 2015
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EXAMPLE
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a(6) = 1 since 2*3 is the only number of the form p*q less than or equal to 6.
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MAPLE
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f:=proc(n) local c, i, j, p, q; c:=0; for i from 1 to n do p:=ithprime(i); if p^2 >= n then break; fi; for j from i+1 to n do q:=ithprime(j); if p*q > n then break; fi; c:=c+1; od: od; RETURN(c); end; # N. J. A. Sloane, Sep 10 2008
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MATHEMATICA
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fPi[n_] := Sum[ PrimePi[n/ Prime@i] - i, {i, PrimePi@ Sqrt@ n}]; Array[ fPi, 81] (* Robert G. Wilson v, Jul 22 2008 *)
Accumulate[Table[If[PrimeOmega[n] MoebiusMu[n]^2 == 2, 1, 0], {n, 100}]] (* Wesley Ivan Hurt, Jun 01 2017 *)
Accumulate[Table[If[SquareFreeQ[n]&&PrimeOmega[n]==2, 1, 0], {n, 100}]] (* Harvey P. Dale, Aug 05 2019 *)
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PROG
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(PARI) a(n)=sum(k=1, n, if(abs(omega(k)-2)+(1-issquarefree(k)), 0, 1))
(PARI) a(n) = my(t=0, i=0); forprime(p = 2, sqrtint(n), i++; t+=primepi(n\p)); t-binomial(i+1, 2) \\ David A. Corneth, Jun 02 2017
(PARI) upto(n) = {my(l=List(), res=[0, 0, 0, 0, 0], j=1, t=0); forprime(p = 2, n, forprime(q=nextprime(p+1), n\p, listput(l, p*q))); listsort(l); for(i=2, #l, t++; res=concat(res, vector(l[i]-l[i-1], j, t))); res} \\ David A. Corneth, Jun 02 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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