A072000 -id:A072000 - cp's OEIS frontend

A072114 Number of 3-almost primes (A014612) <= n.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 9, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 19, 19, 19, 19, 19, 19, 19

Offset: 0

Benoit Cloitre, Jun 19 2002

Number of k <= n such that bigomega(k) = 3.
Let A be a positive integer then card{ x <= n : bigomega(x) = A } ~ (n/Log(n))*Log(Log(n))^(A-1)/(A-1)!. For which n, card{ x <= n : bigomega(x) = 3 } >= card{ x <= n : bigomega(x) = 2 } ?
15530 is the first number for which there are more 3-almost primes than 2-almost primes. See A125149.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig; third edition : Chelsea, New York (1974).
G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A014612, A109251, A001358, A072000.
Partial sums of A101605.
Cf. A125149.

Table[Sum[KroneckerDelta[PrimeOmega[i], 3], {i, n}], {n, 0, 50}] (* Wesley Ivan Hurt, Oct 07 2014 *)

for(n=1,100,print1(sum(i=1,n,bigomega(i)==3),","))

a(n)=my(j,s);forprime(p=2,(n+.5)^(1/3),j=primepi(p)-2;forprime(q=p,sqrtint(n\p),s+=primepi(n\(p*q))-j++));s \\ Charles R Greathouse IV, Mar 21 2012

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A072114(n): return int(sum(primepi(n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(n,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(n//k)+1),a))) # Chai Wah Wu, Aug 17 2024

a(n) = card{ x <= n : bigomega(x) = 3 }, asymptotically : a(n) ~ (n/log(n))*log(log(n))^2/2 [Landau, p. 211].

A082997 a(n) = card{ x <= n : omega(x) = 2 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 15, 16, 16, 17, 18, 19, 19, 19, 19, 20, 21, 22, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 31, 31, 31, 32, 33, 33, 34, 34, 34, 35, 36, 36, 36, 37, 37, 38, 39, 40, 41

Offset: 1

Benoit Cloitre, May 30 2003

nonn

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001221, A007774, A025528, A072000, A082998, A346621.

Partial sums of A215480.

a:= proc(n) option remember; `if`(n=0, 0,
      a(n-1)+`if`(nops(ifactors(n)[2])=2, 1, 0))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 23 2021

a[n_] := Count[PrimeNu[Range[n]], 2];
Array[a, 100] (* Jean-François Alcover, Mar 02 2022 *)

PARI
```
a(n)=sum(i=1,n,if(omega(i)-2,0,1))
```

a(n) = my(s = sqrtint(n), p = 2, j = 1, count = 0); while(p <= s, my(r = nextprime(p+1)); my(t = p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021

from sympy import factorint
from itertools import accumulate
def cond(n): return int(len(factorint(n))==2)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(77)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (n/log(n))*log(log(n)).

a(A007774(n)) = n. - Daniel Suteu, Jul 21 2021

A100951 Number of ways to write n = p*q + r with three distinct primes p, q and r.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 2, 0, 1, 1, 3, 0, 3, 2, 3, 1, 4, 0, 2, 2, 3, 2, 3, 0, 5, 4, 3, 2, 5, 0, 4, 3, 5, 4, 3, 1, 4, 3, 5, 4, 6, 1, 3, 4, 6, 6, 4, 0, 3, 7, 6, 5, 6, 1, 6, 6, 6, 4, 4, 3, 4, 7, 7, 6, 6, 2, 6, 7, 8, 8, 3, 2, 7, 6, 7, 8, 8, 1, 4, 7, 10, 10, 6, 6, 7, 10, 9, 8, 5, 5, 5, 8

Offset: 1

Reinhard Zumkeller, Nov 23 2004

nonn

a(n)<=A100950(n)<=A100949(n)<=Min{A000720(n), A072000(n)}.
a(n) = A000040(i) + A006881(j) for some i and j, such that A000040(i) is not a factor of A006881(j);
a(A100952(n)) = 0.
a(A160373(n)) = n and a(m) <> n for m < A160373(n). [From Reinhard Zumkeller, May 11 2009]

			A100949(21) = #{7+2*7, 11+2*5, 17+2*2} = 3,
a(21) = #{11+2*5} = 1.

R. Zumkeller, Table of n, a(n) for n = 1..10000

A072613 Number of numbers of the form p*q (p, q distinct primes) less than or equal to n.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Aug 11 2002

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

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

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

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A072000.

Partial sums of A280710.

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

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 *)

a(n)=sum(k=1,n,if(abs(omega(k)-2)+(1-issquarefree(k)),0,1))

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

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

from math import isqrt
from sympy import prime, primepi
def A072613(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(1,primepi(isqrt(n))+1))) - primepi(isqrt(n)) # Chai Wah Wu, Jul 23 2024

a(n) = Sum_{pA000720, 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

A205726 Number of semiprimes <= n^2.

Original entry on oeis.org

0, 1, 3, 6, 9, 13, 17, 22, 26, 34, 40, 48, 56, 62, 75, 82, 90, 103, 114, 126, 135, 149, 164, 179, 190, 202, 220, 236, 253, 270, 289, 304, 320, 340, 360, 381, 404, 425, 443, 462, 484, 508, 533, 556, 581, 604, 634, 655, 678, 709, 738, 761, 783, 813, 846, 881

Offset: 1

Keith Backman, Jan 30 2012

nonn

See A205727 and A205728 for related sequences and relationship to Goldbach conjecture.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A038108, A205727, A205728.

SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; nn = 100;  t = Select[Range[nn^2], SemiPrimeQ]; Table[Length[Select[t, # <= n^2 &]], {n, nn}] (* T. D. Noe, Jan 30 2012 *)
Module[{nn=60,sp},sp=Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,nn^2}]];Table[sp[[i^2]],{i,nn}]] (* Harvey P. Dale, May 29 2014 *)

from sympy import prime, primepi
def A205726(n): return int(sum(primepi(n**2//prime(k))-k+1 for k in range(1,primepi(n)+1))) # Chai Wah Wu, Jul 23 2024

a(n) = A072000(A000290(n)). - Michel Marcus, Sep 02 2013

A243906 (Number of semiprimes <= n) - (number of primes <= n).

Original entry on oeis.org

0, -1, -2, -1, -2, -1, -2, -2, -1, 0, -1, -1, -2, -1, 0, 0, -1, -1, -2, -2, -1, 0, -1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 1, 2, 2, 1, 2, 3, 3, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 5, 4, 4, 3, 4, 4, 4, 5, 5, 4, 4, 5, 5, 4, 4, 3, 4, 4, 4, 5, 5, 4, 4, 4, 5, 4, 4, 5, 6, 7, 7, 6, 6, 7, 7, 8

Offset: 1

N. J. A. Sloane, Jun 14 2014

sign

We know from the asymptotic formulas (see Landau) that the sequence is almost always positive.

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; third edition : Chelsea, New York (1974).

N. J. A. Sloane, Table of n, a(n) for n = 1..5195

Cf. A243903, A276176.

Cf. A000720, A072000.

g:= proc(n) if isprime(n) then -1 elif numtheory:-bigomega(n) = 2 then 1 else 0 fi end proc:
ListTools:-PartialSums(map(g, [$1..100])); # Robert Israel, Dec 20 2022

Accumulate[Table[Which[PrimeQ[n],-1,PrimeOmega[n]==2,1,True,0],{n,1000}]] (* Harvey P. Dale, Jun 15 2014 *)

a(n) = #select(x->(bigomega(x) == 2), [1..n]) - primepi(n); \\ Michel Marcus, Dec 20 2022

from math import isqrt
from sympy import prime, primepi
def A243906(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(1,primepi(isqrt(n))+1)))-primepi(n) # Chai Wah Wu, Jul 23 2024

a(n) = A072000(n) - A000720(n). - Michel Marcus, Dec 20 2022

A082996 a(n) = card{ x <= n : bigomega(x) = 4 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Benoit Cloitre, May 30 2003

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A072000, A072114.

Partial sums of A101637.

PARI
```
a(n)=sum(i=1,n,bigomega(i)==4)
```

a(n)=my(j,s);forprime(p=2,(n+.5)^(1/4),forprime(q=p,(n/p+.5)^(1/3),j=primepi(q)-2;forprime(r=q,sqrtint(n\(p*q)),s+=primepi(n\(p*q*r))-j++)));s \\ Charles R Greathouse IV, Mar 21 2012

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A082996(n): return int(sum(primepi(n//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(n,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(n//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(n//(k*m))+1),b))) # Chai Wah Wu, Mar 29 2025

a(n) ~ (1/6)*(n/log(n))*log(log(n))^3.

A082998 a(n) = card{ x <= n : omega(x) = 3 }.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 10

Offset: 1

Benoit Cloitre, May 30 2003

nonn

G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, p. 203, Publications de l'Institut Cartan, 1990.

Daniel Suteu, Table of n, a(n) for n = 1..10000

Cf. A001221, A025528, A033992, A072000, A072114, A082997.

PARI
```
a(n)=sum(i=1,n,if(omega(i)-3,0,1))
```

a(n, k = 3, m = 1, p = 2, s = sqrtnint(n\m, k), j = 1) = my(count = 0); if (k==2, while(p <= s, my(r = nextprime(p+1)); my(t = m*p); while (t <= n, my(w = n\t); if(r > w, break); count += primepi(w) - j; my(r2 = r); while(r2 <= w, my(u = t*r2*r2); if(u > n, break); while (u <= n, count += 1; u *= r2); r2 = nextprime(r2+1)); t *= p); p = r; j += 1); return(count)); while(p <= s, my(r = nextprime(p+1)); my(t = m*p); while(t <= n, my(s = sqrtnint(n\t, k-1)); if(r > s, break); count += a(n, k-1, t, r, s, j+1); t *= p); p = r; j += 1); count; \\ Daniel Suteu, Jul 21 2021

from sympy import factorint
from itertools import accumulate
def cond(n): return int(len(factorint(n))==3)
def aupto(nn): return list(accumulate(map(cond, range(1, nn+1))))
print(aupto(105)) # Michael S. Branicky, Jul 21 2021

a(n) ~ (1/2)*(n/log(n))*log(log(n))^2.

a(A033992(n)) = n. - Daniel Suteu, Jul 21 2021

A100950 Number of partitions of n into a coprime pair of a prime and a semiprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 0, 2, 1, 1, 1, 5, 0, 2, 1, 2, 2, 4, 0, 3, 3, 5, 3, 4, 0, 2, 4, 4, 2, 4, 1, 5, 6, 3, 3, 6, 1, 4, 4, 6, 5, 4, 2, 4, 4, 7, 6, 6, 3, 3, 6, 7, 6, 4, 1, 3, 10, 7, 5, 7, 3, 6, 9, 6, 5, 5, 5, 4, 7, 8, 7, 7, 4, 6, 9, 8, 9, 4, 3, 7, 8, 8, 9, 8, 2, 4, 10, 11, 10, 6, 8, 7, 12, 9, 8, 6, 6, 5

Offset: 1

Reinhard Zumkeller, Nov 23 2004

nonn

A100951(n)<=a(n)<=A100949(n)<=Min{A000720(n), A072000(n)}.

a(n) = A000040(i) + A001358(j) for some i and j, such that A000040(i) is not a factor of A001358(j).

			A100949(21) = #{7+2*7, 11+2*5, 17+2*2} = 3,
a(21) = #{11+2*5, 17+2*2} = 2.

A146167 Number of odd squarefree semiprimes (A046388) <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11

Offset: 1

Washington Bomfim, Oct 27 2008

nonn

A346622 is a different although very similar sequence. - N. J. A. Sloane, Aug 23 2021

			a(33)= 3. The semiprimes <=33 are 15, 21 and 33. Formula gives 11-pi(5)-pi(16)+1 = 3.

Cf. A046388, A001358 (semiprimes), A072000 (Number of semiprimes <= n), A000720 (pi(n), the number of primes <= n).

Cf. also A346622.

Accumulate[Table[If[OddQ[n]&&SquareFreeQ[n]&&PrimeOmega[n]==2,1,0],{n,0,100}]] (* Harvey P. Dale, Feb 08 2016 *)

from math import isqrt
from sympy import prime, primepi
def A146167(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(2,primepi(isqrt(n))+1)))-primepi(isqrt(n))+1 if n>3 else 0 # Chai Wah Wu, Jul 23 2024

a(n) = A072000(n) - A000720(floor(sqrt(n))) - A000720(floor(n/2)) + 1.

cp's OEIS Frontend

A072114 Number of 3-almost primes (A014612) <= n.

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A082997 a(n) = card{ x <= n : omega(x) = 2 }.

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A100951 Number of ways to write n = p*q + r with three distinct primes p, q and r.

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A072613 Number of numbers of the form p*q (p, q distinct primes) less than or equal to n.

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A205726 Number of semiprimes <= n^2.

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A243906 (Number of semiprimes <= n) - (number of primes <= n).

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A082996 a(n) = card{ x <= n : bigomega(x) = 4 }.

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