A072114 -id:A072114 - cp's OEIS frontend

A109251 Number of numbers up to 10^n which are products of three primes.

0, 1, 22, 247, 2569, 25556, 250853, 2444359, 23727305, 229924367, 2227121996, 21578747909, 209214982913, 2030133769624, 19717814526785, 191693417109381, 1865380637252270, 18168907486812690, 177123437184971927, 1728190923820610000

Offset: 0

Martin Raab, Aug 19 2005

nonn

			There are 22 numbers with three prime factors up to 10^2: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99.

Cf. A014612 = numbers with three prime factors, A036352 = number of numbers up to 10^n which are products of two primes, A072114.

ThreeAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@ Sqrt[n/Prime@i]}]; Table[ ThreeAlmostPrimePi[10^n], {n, 0, 14}] (* Robert G. Wilson v *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A109251(n):
    r = 10**n
    return int(sum(primepi(r//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(r,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(r//k)+1),a))) # Chai Wah Wu, Sep 18 2024

a(n) = A072114(10^n). - R. J. Mathar, May 25 2008

a(10)-a(14) from Robert G. Wilson v, Feb 06 2006
a(15)-a(17) from Hiroaki Yamanouchi, Aug 30 2014
a(18)-a(19) from Henri Lifchitz, Dec 01 2014

A120034 Number of 3-almost primes t such that 2^n < t <= 2^(n+1).

Original entry on oeis.org

0, 0, 1, 1, 5, 6, 17, 30, 65, 131, 257, 536, 1033, 2132, 4187, 8370, 16656, 33123, 65855, 130460, 259431, 513737, 1019223, 2019783, 4003071, 7930375, 15712418, 31126184, 61654062, 122137206, 241920724, 479226157, 949313939, 1880589368, 3725662783

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 20 2006

nonn

The partial sum equals the number of Pi_3(2^n) = A127396(n).

			(2^3, 2^4] there is one semiprime, namely 12. 8 was counted in the previous entry.

Cf. A014612, A072114, A109251, A036378, A120033 - A120043.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@Sqrt[n/Prime@i]}]; t = Table[ ThreePrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

A127396 Number of 3-almostprimes <= 2^n.

Original entry on oeis.org

0, 0, 1, 2, 7, 13, 30, 60, 125, 256, 513, 1049, 2082, 4214, 8401, 16771, 33427, 66550, 132405, 262865, 522296, 1036033, 2055256, 4075039, 8078110, 16008485, 31720903, 62847087, 124501149, 246638355, 488559079, 967785236, 1917099175, 3797688543

Offset: 1

Robert G. Wilson v, Dec 29 2006

Robert G. Wilson v, Table of n, a(n) for n = 1..53
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A126279, A109251, A125527.

ThreeAlmostPrimePi[n_] := Sum[PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi[Sqrt[n/Prime@i]]}]; Table[ ThreeAlmostPrimePi[2^n], {n, 30}]

a(n) = A072114(2^n). - R. J. Mathar, Aug 26 2011

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

cp's OEIS Frontend

A109251 Number of numbers up to 10^n which are products of three primes.

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A120034 Number of 3-almost primes t such that 2^n < t <= 2^(n+1).

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A127396 Number of 3-almostprimes <= 2^n.

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

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

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