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A072613 Number of numbers of the form p*q (p, q distinct primes) less than or equal to n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

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

REFERENCES

G. Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. (Cambridge Studies in Advanced Mathematics 1995.)

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Semiprime.

FORMULA

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

a(n) = Sum_{i<=n | tau(i)=4} mu(i). - Wesley Ivan Hurt, May 25 2015

EXAMPLE

a(6) = 1 since 2*3 is the only number of the form p*q less than or equal to 6.

MAPLE

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

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A072000.

Sequence in context: A243283 A243284 A338623 * A029551 A171482 A132015

Adjacent sequences:  A072610 A072611 A072612 * A072614 A072615 A072616

KEYWORD

easy,nonn

AUTHOR

Benoit Cloitre, Aug 11 2002

STATUS

approved

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Last modified September 20 10:18 EDT 2021. Contains 347584 sequences. (Running on oeis4.)