A014613 -id:A014613 - cp's OEIS frontend

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

nonn

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

			a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1-almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..228 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[ Array[a,i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n-1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A078841(n):
    if n <= 1: return (n<<1)+1
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Conjecture: Lim as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007

a(14)-a(29) from Robert G. Wilson v, Feb 11 2006

A114106 Number of 4-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 12, 149, 1712, 18744, 198062, 2050696, 20959322, 212385942, 2139236881, 21454599814, 214499908019, 2139634739326, 21306682904040, 211905511283590, 2105504493045818, 20905484578206982, 207458897819329031, 2057931819347474462

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 12 primes with four almost primes up to 100: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90 and 100, so a(2) = 12.

Cf. A006880, A066265, A109251, A014613.

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; Table[ FourAlmostPrimePi[n], {n, 0, 13}]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A114106(n): return sum(primepi(10**n//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(10**n,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(10**n//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(10**n//(k*m))+1),b)) # Chai Wah Wu, Aug 17 2024

a(14) from Robert G. Wilson v, Jan 07 2007
a(15)-a(17) from Henri Lifchitz, Jul 21 2015
a(18)-a(19) from Henri Lifchitz, Feb 02 2025

A101695 a(n) = n-th n-almost prime.

Original entry on oeis.org

2, 6, 18, 40, 108, 224, 480, 1296, 2688, 5632, 11520, 25600, 53248, 124416, 258048, 540672, 1105920, 2228224, 4587520, 9830400, 19922944, 40894464, 95551488, 192937984, 396361728, 822083584, 1660944384, 3397386240, 6845104128

Offset: 1

Jonathan Vos Post, Dec 12 2004

nonn

A k-almost-prime is a positive integer that has exactly k prime factors, counted with multiplicity.

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. This is the diagonal just below A078841.

			a(1) = first 1-almost prime = first prime = A000040(1) = 2.
a(2) = 2nd 2-almost prime = 2nd semiprime = A001358(2) = 6.
a(3) = 3rd 3-almost prime = A014612(3) = 18.
a(4) = 4th 4-almost prime = A014613(4) = 40.
a(5) = 5th 5-almost prime = A014614(5) = 108.

Charles R Greathouse IV, Table of n, a(n) for n = 1..1000 (first 229 terms from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A000040, A001358, A014612, A014613, A046314, A046306, A046308, A046310, A046312, A046314, A069272, A069273, A069274, A069275, A069276, A069277, A069278, A069279, A069280, A069281, A101637, A101638, A101605, A101606.

A101695 := proc(n)
    local s,a ;
    s := 0 ;
    for a from 2^n do
        if numtheory[bigomega](a) = n then
            s := s+1 ;
            if s = n then
                return a;
            end if;
        end if;
    end do:
end proc: # R. J. Mathar, Aug 09 2012

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
AlmostPrime[k_, n_] := Block[{e = Floor[ Log[2, n] + k], a, b}, a = 2^e; Do[b = 2^p; While[ AlmostPrimePi[k, a] < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; AlmostPrime[1, 1] = 2; lst = {}; Do[ AppendTo[lst, AlmostPrime[n, n]], {n, 30}]; lst (* Robert G. Wilson v, Oct 07 2007 *)

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A101695(n):
    if n == 1: return 2
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Conjecture: lim_{ n->inf.} a(n+1)/a(n) = 2. - Robert G. Wilson v, Oct 07 2007, Nov 13 2007

Stronger conjecture: a(n)/(n * 2^n) is polylogarithmic in n. That is, there exist real numbers b < c such that (log n)^b < a(n)/(n * 2^n) < (log n)^c for large enough n. Probably b and c can be chosen close to 0. - Charles R Greathouse IV, Aug 28 2012

a(21)-a(30) from Robert G. Wilson v, Feb 11 2006

a(12) corrected by N. J. A. Sloane, Nov 23 2007

A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 3, 4, 5, 2, 6, 5, 6, 1, 7, 3, 8, 4, 7, 8, 9, 2, 9, 10, 5, 6, 10, 7, 11, 1, 11, 12, 13, 3, 12, 14, 15, 4, 13, 8, 14, 9, 10, 16, 15, 2, 17, 11, 18, 12, 16, 5, 19, 6, 20, 21, 17, 7, 18, 22, 13, 1, 23, 14, 19, 15, 24, 16, 20, 3, 21, 25, 17, 18, 26, 19, 22, 4, 8, 27, 23

Offset: 1

Naohiro Nomoto, Jan 11 2001

Equivalently, the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - Gus Wiseman, Dec 28 2018

There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - Peter Munn, Aug 04 2019

			3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2     3  4  5     6  9   7   8   11  10  14      13  12  17  18
              2     3     4  6   5       7   9   10      11  8   13  12
                    2        4   3       5   6   9       7       11  8
                                 2       3   4   6       5       7
                                         2       4       3       5
                                                         2       3
                                                                 2
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of 1's are A000079.
Cf. A001358, A014612, A014613, A014614.
Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A078843, A209934, A302242, A322838, A322839, A322840.
Equivalent sequence restricted to squarefree numbers: A340313.

p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[bigomega](n);
      p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015

p[] = 0; a[n] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)

a(n) = my(k=bigomega(n)); sum(i=1, n, bigomega(i)==k); \\ Michel Marcus, Jun 27 2024

from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A058933(n):
    if n==1: return 1
    if isprime(n): return primepi(n)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))) # Chai Wah Wu, Aug 28 2024

Ordinal transform of A001222 (bigomega). - Franklin T. Adams-Watters, Aug 28 2006

If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - Peter Munn, Aug 05 2019

Name edited by Peter Munn, Dec 30 2022

A074969 Numbers with six distinct prime divisors.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 60060, 62790, 66990, 67830, 71610, 72930, 78540, 79170, 81510, 82110, 84630, 85470, 87780, 90090, 91770, 92820, 94710, 98670, 99330, 101010, 102102, 103530, 103740, 106260, 106590, 108570

Offset: 1

Zak Seidov, Oct 04 2002

nonn

The smallest number with six distinct prime divisors is the product of the first six primes, 2*3*5*7*11 = 30030.

The smallest number with seven distinct prime divisors is the product of the first seven primes, 2*3*5*7*11*13 = 390390.

			60060 is a term because 60060 = 2^2*3*5*7*11*13 with six distinct prime divisors 2, 3, 5, 7, 11, 13
87780 is a term because 87780 = 2^2*3*5*7*11*19 with six distinct prime divisors 2, 3, 5, 7, 11, 19.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row 6 of A125666.

Cf. A067885, A001358, A014612, A014613, A014614.

Select[Range[0,5*8! ],Length[FactorInteger[ # ]]==6&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)

is(n)=omega(n)==6 \\ Charles R Greathouse IV, Jun 19 2016

A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)
list(lim,pr=6)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

{n : A001221(n) = 6} . - R. J. Mathar, Jul 07 2012

A110227 4-almost primes p * q * r * s relatively prime to p + q + r + s.

Original entry on oeis.org

40, 54, 56, 88, 90, 104, 135, 136, 152, 184, 189, 198, 210, 225, 232, 248, 250, 294, 296, 297, 306, 328, 344, 350, 351, 376, 390, 414, 424, 441, 459, 462, 472, 488, 513, 522, 536, 546, 550, 568, 570, 584, 621, 632, 664, 686, 712, 714, 735, 738, 765, 776

Offset: 1

Jonathan Vos Post, Jul 16 2005

p, q, r, s are not necessarily distinct. The converse to this is A110228: 4-almost primes p * q * r * s not relatively prime to p+q+r+s.

			104 is in this sequence because 104 = 2^3 * 13, which is relatively prime to 2 + 2 + 2 + 13 = 19, which is prime.

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

Cf. A014613, A110187, A110188, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2,lim\8, forprime(q=2,min(p,lim\4\p), my(pq=p*q); forprime(r=2,min(lim\pq\2,q), my(pqr=pq*r,t); forprime(s=2,min(lim\pqr,r), t=pqr*s; if(gcd(t,p+q+r+s)==1, listput(v,t)))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Corrected and extended by Ray Chandler, Jul 20 2005

A110228 4-almost primes p * q * r * s not relatively prime to p + q + r + s.

Original entry on oeis.org

16, 24, 36, 60, 81, 84, 100, 126, 132, 140, 150, 156, 196, 204, 220, 228, 234, 260, 276, 308, 315, 330, 340, 342, 348, 364, 372, 375, 380, 444, 460, 476, 484, 490, 492, 495, 510, 516, 525, 532, 558, 564, 572, 580, 585, 620, 625, 636, 644, 650, 666, 676, 690

Offset: 1

Jonathan Vos Post, Jul 16 2005

p, q, r, s are not necessarily distinct. The converse to this is A110227: 4-almost primes p * q * r * s which are relatively prime to p+q+r+s.

			84 is in this sequence because 84 = 2^2 * 3 * 7 and the sum of these prime factors is 2 + 2 + 3 + 7 = 14 = 2 * 7, which is a divisor of 84.

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

Cf. A014613, A110187, A110188, A110227, A110229, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2,lim\8, forprime(q=2,min(p,lim\4\p), my(pq=p*q); forprime(r=2,min(lim\pq\2,q), my(pqr=pq*r,t); forprime(s=2,min(lim\pqr,r), t=pqr*s; if(gcd(t,p+q+r+s)>1, listput(v,t)))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Corrected and extended by Ray Chandler, Jul 20 2005

A120049 Number of 8-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012, 291646797, 3173159326, 34192782745, 365561221293, 3882841742380, 41015564702074, 431227959019552, 4515480975731045, 47115876816676830

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 7 eight-almost primes up to 1000: 256, 384, 576, 640, 864, 896 & 960.

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[8, 10^n], {n, 12}]

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120049(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,8))) # Chai Wah Wu, Aug 23 2024

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Aug 13 2018
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

Original entry on oeis.org

13, 15, 117, 126, 270, 2576, 8820, 16560, 21168, 46848, 295245, 441600, 846720, 4078080, 80663040, 40590720, 2173236480, 4011724800, 21122906112, 40915058688, 274148425728, 63769149440, 2707602702336, 6167442456576, 21586195906560, 29798871072768, 420127895977984, 631722992467968

Offset: 1

Jonathan Vos Post, Jun 16 2005

An emirp ("prime" spelled backwards) is a prime whose (base 10) reversal is also prime, but which is not a palindromic prime. The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (A006567).
An emirpimes ("semiprime" spelled backwards) is a semiprime whose (base 10) reversal is a different semiprime. A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (A097393).
An "emirp tsomla-3" ("3-almost prime" spelled backwards) is the k=3 sequence of the series for which k=1 are emirps and k=2 are emirpimes, a list of these being A109023. The union of these for k=1 through k = 13 is A109019.
The primes correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (A000040). The 2-almost prime numbers correspond to semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... (A001358).
The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... (A014613).
The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).
The Mathematica code for this was written by Ray Chandler, who has coauthorship credit for this sequence.

			a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.
a(2) = 15 because 15 is the smallest emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, since 15 reversed is 51 and both 15 and 51 are semiprimes (i.e. 15 = 3 * 5 and 51 = 3 * 17).
a(3) = 117 because 117 is the smallest "emirp tsomla-3" ("3-almost prime" spelled backwards) since 117 reversed is 711 and 117 = 3^2 * 13 and 711 = 3^2 * 79.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

kAlmost[n_] := Plus @@ Last /@ FactorInteger@n; fQ[n_] := Block[{id = IntegerDigits@n, k = kAlmost@n}, If[id != Reverse@id && k == kAlmost@FromDigits@Reverse@id, k, -1]]; t = Table[0, {20}]; Do[ a = fQ@n; If[a < 20 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 10, 150000000}] (* Robert G. Wilson v, Jan 06 2008 *)
Table[Select[Range[41*10^5],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==n&][[1]],{n,14}] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Oct 15 2023 *)

a(14)-a(16) from Robert G. Wilson v, Jan 06 2008
a(17)-a(24) from Donovan Johnson, Nov 17 2008
a(25)-a(28) from Michael S. Branicky, Jun 04 2024

A120047 Number of 6-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 2, 37, 485, 5933, 68963, 774078, 8493366, 91683887, 977694273, 10327249593, 108264085934, 1128049914377, 11694704489580, 120734708167792, 1242063105505230, 12739510126065301, 130330025583399801

Offset: 0

Robert G. Wilson v, Feb 07 2006

			There are 2 six-almost primes up to 100: 64 and 96, so a(2) = 2.

Cf. A066265, A014613, A116382.
Cf. A006880, A036352, A109251, A114106, A114453.
Cf. A120048, A120049, A120050, A120051, A120052, A120053, A116430.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[6, 10^n], {n, 0, 13}]

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def almostprimepi(n,k):
    if k==0: return int(n>=1)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n))
def A120047(n): return almostprimepi(10**n,6) # Chai Wah Wu, Dec 09 2024

a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(18) from Henri Lifchitz, Feb 03 2025

cp's OEIS Frontend

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

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A114106 Number of 4-almost primes less than or equal to 10^n.

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A101695 a(n) = n-th n-almost prime.

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A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

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A074969 Numbers with six distinct prime divisors.

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A110227 4-almost primes p * q * r * s relatively prime to p + q + r + s.

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A110228 4-almost primes p * q * r * s not relatively prime to p + q + r + s.

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