A078840 -id:A078840 - cp's OEIS frontend

A069281 20-almost primes (generalization of semiprimes).

1048576, 1572864, 2359296, 2621440, 3538944, 3670016, 3932160, 5308416, 5505024, 5767168, 5898240, 6553600, 6815744, 7962624, 8257536, 8650752, 8847360, 8912896, 9175040, 9830400, 9961472, 10223616, 11943936, 12058624

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 20 not necessarily distinct primes.
Divisible by exactly 20 prime powers (not including 1).
Any 20-almost prime can be represented in several ways as a product of two 10-almost primes A046314; in several ways as a product of four 5-almost primes A014614; in several ways as a product of five 4-almost primes A014613; and in several ways as a product of ten semiprimes A001358. - Jonathan Vos Post, Dec 12 2004

D. W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A101637, A101638, A101605, A101606.

Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), this sequence (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[2*9!,5*10! ],Plus@@Last/@FactorInteger[ # ]==20 &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

k=20; start=2^k; finish=15000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v \\ Depending upon the size of k and how many terms are needed, a much more efficient algorithm than the brute-force method above may be desirable. See additional comments in this section of A069280.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069281(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)))
    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,0,1,1,20)))
    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

Product p_i^e_i with Sum e_i = 20.

a(n) = A078840(20,n). - R. J. Mathar, Jan 30 2019

A078843 Where 3^n occurs in n-almost primes, starting at a(0)=1.

Original entry on oeis.org

1, 2, 3, 5, 8, 14, 23, 39, 64, 103, 169, 269, 427, 676, 1065, 1669, 2628, 4104, 6414, 10023, 15608, 24281, 37733, 58503, 90616, 140187, 216625, 334527, 516126, 795632, 1225641, 1886570, 2901796, 4460359, 6851532, 10518476, 16138642, 24748319

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

nonn

			a(3) = 5 since 3^3 is the 5th 3-almost-prime: 8,12,18,20,27,....., A014612.

Max Alekseyev, Table of n, a(n) for n = 0..50
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078844, A078845, A078846.

AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 3^n], {n, 2, 37}] (* Robert G. Wilson v, Feb 09 2006 *)

a(n)=sum(i=1,3^n,if(bigomega(i)-n,0,1))

{ appi(k,n,m=2) = local(r=0);
if(k==0,return(1));
if(k==1,return(primepi(n)));
forprime(p=m, floor(sqrtn(n,k)+1e-20),
r+=appi(k-1,n\p,p)-(k==2)*(primepi(p)-1));
r }
{ appi3(k,n) = appi(k,n) - if(k>=1,appi(k-1,n\3)) }
a=1; for(n=1,50, k=ceil(n*log(5/3)/log(5/2)); a+=appi3(n-k,3^n\2^k); print1(a,", "))
\\ Max Alekseyev, Jan 06 2008

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A078843(n):
    def almostprimepi(n,k):
        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))
    return almostprimepi(3**n,n) if n else 1 # Chai Wah Wu, Sep 01 2024

a(n) = a(n-1) + appi3(n-k, floor(3^n/2^k)), where k = ceiling(n*c) with c = log(5/3)/log(5/2) = 0.55749295065024006729857073190835923443... and appi3(k,n) is the number of k-almost primes not divisible by 3 and not exceeding n. - Max Alekseyev, Jan 06 2008

a(14)-a(37) from Robert G. Wilson v, Feb 09 2006

A078844 Where 5^n occurs in n-almost-primes, starting at a(0)=1.

Original entry on oeis.org

1, 3, 9, 30, 90, 269, 788, 2249, 6340, 17526, 47911, 129639, 348251, 929714, 2469499, 6532869, 17219031, 45246630, 118572805, 309998131, 808746993, 2105893899, 5474080107, 14207001052, 36818679828, 95292132897, 246327403310

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(2) = 9 since 5^2 is the 9th 2-almost-prime: {4,6,9,10,14,15,21,22,25,...}.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078843, A078845, A078846.

l = Table[0, {30}]; e = 0; Do[f = Plus @@ Last /@ FactorInteger[n]; l[[f+1]]++; If[n == 5^e, Print[l[[f+1]]]; e++ ], {n, 1, 5^10}] (* Ryan Propper, Aug 08 2005 *)
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 *)
Join[{1},Table[ AlmostPrimePi[n, 5^n], {n, 1, 25}]] (* Robert G. Wilson v, Feb 10 2006 *)

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 A078844(n): return almostprimepi(5**n, n) if n else 1 # Chai Wah Wu, Nov 07 2024

a(8)-a(10) from Ryan Propper, Aug 08 2005
a(11)-a(25) from Robert G. Wilson v, Feb 10 2006
a(26) from Donovan Johnson, Sep 27 2010

A078846 Where 11^n occurs in n-almost-primes, starting at a(0)=1.

Original entry on oeis.org

1, 5, 40, 328, 2556, 18452, 126096, 827901, 5276913, 32887213, 201443165, 1217389949, 7279826998, 43168558912, 254258462459, 1489291941733, 8683388113017, 50433408838966

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

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

			a(2) = 40 since 11^2 is the 40th 2-almost-prime: A001358(40) = 121.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078843, A078844, A078845.

AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 11^n], {n, 2, 11}] (* Robert G. Wilson v, Feb 09 2006 *)

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(11^n, n)); \\ Daniel Suteu, Jul 10 2023

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A078846(n):
    def almostprimepi(n, k):
        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))
    return almostprimepi(11**n, n) if n else 1 # Chai Wah Wu, Sep 01 2024

a(6)-a(11) from Robert G. Wilson v, Feb 09 2006
a(12)-a(15) from Donovan Johnson, Sep 27 2010
a(16)-a(17) from Daniel Suteu, Jul 10 2023

A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 34, 247, 1712, 11185, 68963, 409849, 2367507, 13377156, 74342563, 407818620, 2214357712, 11926066887, 63809981451, 339576381990, 1799025041767, 9494920297227, 49950199374227, 262036734664892

Offset: 0

Robert G. Wilson v, Feb 10 2006, Jun 01 2006

nonn

If instead we asked for those less than or equal to 2^n, then the sequence is A000012.

Cf. A036352, A114106, A114453.
Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.
Cf. A006880, A036352, A109251, A114106, A114453, A120047 - A120053.

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[n, 10^n], {n, 0, 13}]

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(10^n, n)); \\ Daniel Suteu, Jul 10 2023

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A116430(n):
    if n<=1: return 3*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(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n,0,1,1,n))) # Chai Wah Wu, Aug 23 2024

Edited by N. J. A. Sloane, Aug 08 2008 at the suggestion of R. J. Mathar
a(15)-a(16) from Donovan Johnson, Oct 01 2010
a(17)-a(19) from Daniel Suteu, Jul 10 2023

A078845 Where 7^n occurs in n-almost-primes, starting at a(0)=1.

Original entry on oeis.org

1, 4, 17, 82, 385, 1688, 7089, 28893, 115180, 450906, 1740244, 6640747, 25115604, 94312569, 352110321, 1308256678, 4841115048, 17852264639, 65636109307, 240689877440, 880582139867

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(2) = 17 since 7^2 is the 17th 2-almost-prime: {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,...}.

Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078841, A078842, A078843, A078844, A078846.

l = Table[0, {30}]; e = 0; Do[f = Plus @@ Last /@ FactorInteger[n]; l[[f+1]]++; If[n == 7^e, Print[l[[f+1]]]; e++ ], {n, 1, 7^10}] (* Ryan Propper, Aug 08 2005 *)
AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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[n, 7^n], {n, 2, 15}] (* Robert G. Wilson v, Feb 09 2006 *)

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 A078845(n): return almostprimepi(7**n,n) if n else 1 # Chai Wah Wu, Oct 02 2024

a(7)-a(10) from Ryan Propper, Aug 08 2005
a(11)-a(15) from Robert G. Wilson v, Feb 09 2006
a(16)-a(20) from Donovan Johnson, Sep 27 2010

A091538 Triangle built from m-primes as columns.

Original entry on oeis.org

1, 0, 2, 0, 3, 4, 0, 5, 6, 8, 0, 7, 9, 12, 16, 0, 11, 10, 18, 24, 32, 0, 13, 14, 20, 36, 48, 64, 0, 17, 15, 27, 40, 72, 96, 128, 0, 19, 21, 28, 54, 80, 144, 192, 256, 0, 23, 22, 30, 56, 108, 160, 288, 384, 512, 0, 29, 25, 42, 60, 112, 216, 320, 576, 768, 1024

Offset: 0

Wolfdieter Lang, Feb 13 2004

m-primes (also called m-almost primes) are the numbers which have precisely m prime factors counting multiple factors. 1 is included as 0-prime.

The number N>=1 appears in column no. m = A001222(N).

			From _Michael De Vlieger_, May 24 2017: (Start)
Chart a(n,m) read by antidiagonals:
  n | m ->
  ------------------------------------------------
  0 |    1     0     0     0     0     0     0 ... (A000007)
  1 |    2     3     5     7    11    13    17     (A000040)
  2 |    4     6     9    10    14    15    21     (A001358)
  3 |    8    12    18    20    27    28    30     (A014612)
  4 |   16    24    36    40    54    56    60     (A014613)
  5 |   32    48    72    80   108   112   120     (A014614)
  6 |   64    96   144   160   216   224   240     (A046306)
  7 |  128   192   288   320   432   448   480     (A046308)
  8 |  256   384   576   640   864   896   960     (A046310)
       ...
Triangle begins:
  0 |    1
  1 |    0    2
  2 |    0    3    4
  3 |    0    5    6    8
  4 |    0    7    9   12   16
  5 |    0   11   10   18   24   32
  6 |    0   13   14   20   36   48    64
  7 |    0   17   15   27   40   72    96   128
  8 |    0   19   21   28   54   80   144   192   256
       ...
(End)

Chai Wah Wu, Table of n, a(n) for n = 0..10000
Wolfdieter Lang, First 11 rows.

The column sequences (without leading zeros) are: A000007, A000040 (primes), A001358, A014612-4, A046306, A046308, A046310, A046312, A046314, A069272-A069281 for m=0..20, respectively.

A078840 is this table with the zeros omitted.

With[{nn = 11}, Function[s, Function[t, Table[Function[m, If[m == 1, Boole[k == 1], t[[m, k]]]][n - k + 1], {n, nn}, {k, n, 1, -1}]]@ Map[Position[s, #][[All, 1]] &, Range[0, nn]]]@ PrimeOmega@ Range[2^nn]] (* or *)
a = {1}; Do[Block[{r = {Prime@ n}}, Do[AppendTo[r, SelectFirst[ Range[a[[-(n - i)]] + 1, 2^n], PrimeOmega@ # == i &]], {i, 2, n - 1}]; a = Join[a, {0}, If[n == 1, {}, r], {2^n}]], {n, 11}]; a (* Michael De Vlieger, May 24 2017 *)

from math import isqrt, comb, prod
from sympy import prime, primerange, integer_nthroot, primepi
def A091538(n):
    a = (m:=isqrt(k:=n+1<<1))+(k>m*(m+1))
    r = n-comb(a,2)
    w = a-r
    if r==0: return int(w==1)
    if r==1: return prime(w)
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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(w+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,r)))
    return bisection(f,w,w) # Chai Wah Wu, Jun 11 2025

For n>=m>=1: a(n, m)= (n-m+1)-th member in the strictly monotonically increasing sequence of numbers N satisfying: N=product(p(k)^(e_k), k=1..) with p(k) := A000040(k) (k-th prime) such that sum(e_k, k=1..) = m, where the e_k are nonnegative. if m=0 : a(n, 0)=1 if n=0 else 0. If n

A116426 The number of n-almost primes less than or equal to 4^n, starting with a(0)=1.

Original entry on oeis.org

1, 2, 6, 13, 34, 77, 177, 406, 887, 1962, 4225, 9094, 19482, 41414, 87706, 184976, 389357, 816193, 1708412, 3566209, 7431153, 15457234, 32098652, 66560309, 137830562, 285062028, 588871107, 1215176367, 2505048537, 5159228725

Offset: 0

Robert G. Wilson v, Feb 10 2006

nonn

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

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 *)
Join[{1},Table[AlmostPrimePi[n, 4^n], {n, 29}]]

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116426(n):
    if n<=1: return 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((1<<(n<<1))//prod(c[1] for c in a))-a[-1][0] for a in g(1<<(n<<1),0,1,1,n))) # Chai Wah Wu, Oct 02 2024

A116427 The number of n-almost primes less than or equal to 6^n, starting with a(0)=1.

Original entry on oeis.org

1, 3, 13, 50, 200, 726, 2613, 9061, 30779, 102637, 338230, 1102674, 3566001, 11455355, 36597558, 116395587, 368749900, 1164407829, 3666312894, 11515047829, 36085395700, 112857846859, 352329509934, 1098136237818

Offset: 0

Robert G. Wilson v, Feb 10 2006

nonn

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

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 *)
Join[{1},Table[AlmostPrimePi[n, 6^n], {n, 21}]]

a(22)-a(23) from Donovan Johnson, Oct 01 2010

A116428 The number of n-almost primes less than or equal to 8^n, starting with a(0)=1.

Original entry on oeis.org

1, 4, 22, 125, 669, 3410, 16677, 78369, 359110, 1612613, 7133274, 31185350, 135062165, 580556958, 2480278767, 10542976739, 44626102826, 188215850830, 791374442571, 3318478309647, 13882441625034, 57952990683107

Offset: 0

Robert G. Wilson v, Feb 14 2006

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

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}]]]]];
Table[ AlmostPrimePi[n, 8^n], {n, 14}] (* Eric W. Weisstein, Feb 07 2006 *)

almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(8^n, n)); \\ Daniel Suteu, Jul 10 2023

a(15)-a(18) from Donovan Johnson, Oct 01 2010

a(19)-a(21) from Daniel Suteu, Jul 10 2023

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A078846 Where 11^n occurs in n-almost-primes, starting at a(0)=1.

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A116430 The number of n-almost primes less than or equal to 10^n, starting with a(0)=1.

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A078845 Where 7^n occurs in n-almost-primes, starting at a(0)=1.

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A091538 Triangle built from m-primes as columns.

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