A066265 -id:A066265 - cp's OEIS frontend

A001358 Semiprimes (or biprimes): products of two primes.

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Offset: 1

N. J. A. Sloane, R. K. Guy

Numbers of the form p*q where p and q are primes, not necessarily distinct.
These numbers are sometimes called semiprimes or 2-almost primes.
Numbers n such that Omega(n) = 2 where Omega(n) = A001222(n) is the sum of the exponents in the prime decomposition of n.
Complement of A100959; A064911(a(n)) = 1. - Reinhard Zumkeller, Nov 22 2004
The graph of this sequence appears to be a straight line with slope 4. However, the asymptotic formula shows that the linearity is an illusion and in fact a(n)/n ~ log(n)/log(log(n)) goes to infinity. See also the graph of A066265 = number of semiprimes < 10^n.
For numbers between 33 and 15495, semiprimes are more plentiful than any other k-almost prime. See A125149.
Numbers that are divisible by exactly 2 prime powers (not including 1). - Jason Kimberley, Oct 02 2011
The (disjoint) union of A006881 and A001248. - Jason Kimberley, Nov 11 2015
An equivalent definition of this sequence is a'(n) = smallest composite number which is not divided by any smaller composite number a'(1),...,a'(n-1). - Meir-Simchah Panzer, Jun 22 2016
The above characterization can be simplified to "Composite numbers not divisible by a smaller term." This shows that this is the equivalent of primes computed via Eratosthenes's sieve, but starting with the set of composite numbers (i.e., complement of 1 union primes) instead of all positive integers > 1. It's easy to see that iterating the method (using Eratosthenes's sieve each time on the remaining numbers, complement of the previously computed set) yields numbers with bigomega = k for k = 0, 1, 2, 3, ..., i.e., {1}, A000040, this, A014612, etc. - M. F. Hasler, Apr 24 2019
For all n except n = 2, a(n) is a deficient number. - Amrit Awasthi, Sep 10 2024
It is reasonable to assume that the "comforting numbers" which John T. Williams found in Chapter 3 of Milne's book "The House at Pooh Corner" are these semiprimes. Winnie-the-Pooh wonders whether he has 14 or 15 honey pots and concludes: "It's sort of comforting." To arrange a semiprime number of honey pots in a rectangular way, let's say on a shelf, with the larger divisor parallel to the wall, there is only one solution and this is for a simple mind like Winnie-the-Pooh comforting. - Ruediger Jehn, Dec 12 2024

			From _Gus Wiseman_, May 27 2021: (Start)
The sequence of terms together with their prime factors begins:
   4 = 2*2     46 = 2*23     91 = 7*13    141 = 3*47
   6 = 2*3     49 = 7*7      93 = 3*31    142 = 2*71
   9 = 3*3     51 = 3*17     94 = 2*47    143 = 11*13
  10 = 2*5     55 = 5*11     95 = 5*19    145 = 5*29
  14 = 2*7     57 = 3*19    106 = 2*53    146 = 2*73
  15 = 3*5     58 = 2*29    111 = 3*37    155 = 5*31
  21 = 3*7     62 = 2*31    115 = 5*23    158 = 2*79
  22 = 2*11    65 = 5*13    118 = 2*59    159 = 3*53
  25 = 5*5     69 = 3*23    119 = 7*17    161 = 7*23
  26 = 2*13    74 = 2*37    121 = 11*11   166 = 2*83
  33 = 3*11    77 = 7*11    122 = 2*61    169 = 13*13
  34 = 2*17    82 = 2*41    123 = 3*41    177 = 3*59
  35 = 5*7     85 = 5*17    129 = 3*43    178 = 2*89
  38 = 2*19    86 = 2*43    133 = 7*19    183 = 3*61
  39 = 3*13    87 = 3*29    134 = 2*67    185 = 5*37
(End)

Archimedeans Problems Drive, Eureka, 17 (1954), 8.
Raymond Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter II, Problem 60.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Vol. 1, Teubner, Leipzig; third edition: Chelsea, New York (1974). See p. 211.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
John T. Williams, Pooh and the Philosophers, Dutton Books, 1995.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
Dragos Crisan and Radek Erban, On the counting function of semiprimes, INTEGERS, Vol. 21 (2021), #A122.
Daniel A. Goldston, Sidney W. Graham, János Pintz and Cem Y. Yildirim, Small gaps between primes or almost primes, Transactions of the American Mathematical Society, Vol. 361, No. 10 (2009), pp. 5285-5330, arXiv preprint, arXiv:math/0506067 [math.NT], 2005.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968
Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, alternative link.
Donovan Johnson, Jonathan Vos Post, and Robert G. Wilson v, Selected n and a(n). (2.5 MB)
Dixon Jones, Quickie 593, Mathematics Magazine, Vol. 47, No. 3, May 1974, p. 167.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909. See Vol. 1, p. 211.
Xianmeng Meng, On sums of three integers with a fixed number of prime factors, Journal of Number Theory, Vol. 114, No. 1 (2005), pp. 37-65.
Michael Penn, What makes a number "good"?, YouTube video, 2022.
Carlos Rivera, Conjecture 108: On the counting function of semiprimes, The Prime Puzzles & Problems Connection.
Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Almost Prime.
Wikipedia, Almost prime.
Robert G. Wilson v, Subsequences at various powers of 10.
Index to sequences related to sums of cubes
Index entries for "core" sequences

Cf. A064911 (characteristic function).
Cf. A048623, A048639, A000040 (primes), A014612 (products of 3 primes), A014613, A014614, A072000 ("pi" for semiprimes), A065516 (first differences).
Cf. A077554, A077555, A002024, A072966, A100592, A014673, A068318, A061299, A087718, A089994, A089995, A096916, A096932, A106550, A106554, A108541, A108542, A126663, A131284, A138510, A138511, A072931, A088183, A171963, A237040 (semiprimes of form n^3 + 1).
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r=1), this sequence (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), A069281 (r=20).
These are the Heinz numbers of length-2 partitions, counted by A004526.
The squarefree case is A006881 with odd/even terms A046388/A100484 (except 4).
Including primes gives A037143.
The odd/even terms are A046315/A100484.
Partial sums are A062198.
The prime factors are A084126/A084127.
Grouping by greater factor gives A087112.
The product/sum/difference of prime indices is A087794/A176504/A176506.
Positions of even/odd terms are A115392/A289182.
The terms with relatively prime/divisible prime indices are A300912/A318990.
Factorizations using these terms are counted by A320655.
The prime indices are A338898/A338912/A338913.
Grouping by weight (sum of prime indices) gives A338904, with row sums A024697.
The terms with even/odd weight are A338906/A338907.
The terms with odd/even prime indices are A338910/A338911.
The least/greatest term of weight n is A339114/A339115.
Cf. A014342, A098350, A112141, A320732, A332765, A339003/A339004, A339116.

a001358 n = a001358_list !! (n-1)
a001358_list = filter ((== 2) . a001222) [1..]

[n: n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Bruno Berselli, Sep 09 2015

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a) = 2 then return a; end if; end do: end if; end proc:
seq(A001358(n), n=1..120) ; # R. J. Mathar, Aug 12 2010

Select[Range[200], Plus@@Last/@FactorInteger[#] == 2 &] (* Zak Seidov, Jun 14 2005 *)
Select[Range[200], PrimeOmega[#]==2&] (* Harvey P. Dale, Jul 17 2011 *)

select( isA001358(n)={bigomega(n)==2}, [1..199]) \\ M. F. Hasler, Apr 09 2008; added select() Apr 24 2019

list(lim)=my(v=List(),t);forprime(p=2, sqrt(lim), t=p;forprime(q=p, lim\t, listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Sep 11 2011

A1358=List(4); A001358(n)={while(#A1358M. F. Hasler, Apr 24 2019

from sympy import factorint
def ok(n): return sum(factorint(n).values()) == 2
print([k for k in range(1, 190) if ok(k)]) # Michael S. Branicky, Apr 30 2022

from math import isqrt
from sympy import primepi, prime
def A001358(n):
    def f(x): return int(n+x-sum(primepi(x//prime(k))-k+1 for k in range(1, primepi(isqrt(x))+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Jul 23 2024

a(n) ~ n*log(n)/log(log(n)) as n -> infinity [Landau, p. 211], [Ayoub].
Recurrence: a(1) = 4; for n > 1, a(n) = smallest composite number which is not a multiple of any of the previous terms. - Amarnath Murthy, Nov 10 2002
A174956(a(n)) = n. - Reinhard Zumkeller, Apr 03 2010
a(n) = A088707(n) - 1. - Reinhard Zumkeller, Feb 20 2012
Sum_{n>=1} 1/a(n)^s = (1/2)*(P(s)^2 + P(2*s)), where P is the prime zeta function. - Enrique Pérez Herrero, Jun 24 2012
sigma(a(n)) + phi(a(n)) - mu(a(n)) = 2*a(n) + 1. mu(a(n)) = ceiling(sqrt(a(n))) - floor(sqrt(a(n))). - Wesley Ivan Hurt, May 21 2013
mu(a(n)) = -Omega(a(n)) + omega(a(n)) + 1, where mu is the Moebius function (A008683), Omega is the count of prime factors with repetition, and omega is the count of distinct prime factors. - Alonso del Arte, May 09 2014
a(n) = A078840(2,n). - R. J. Mathar, Jan 30 2019
A100484 UNION A046315. - R. J. Mathar, Apr 19 2023
Conjecture: a(n)/n ~ (log(n)/log(log(n)))*(1-(M/log(log(n)))) as n -> oo, where M is the Mertens's constant (A077761). - Alain Rocchelli, Feb 02 2025

More terms from James Sellers, Aug 22 2000

A072000 Number of semiprimes (A001358) <= n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 13, 13, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 1

Benoit Cloitre, Jun 19 2002

Number of k <= n such that bigomega(k) = 2.

A. Hildebrand, On the number of prime factors of an integer, in Ramanujan Revisited (Urbana-Champaign, Ill., 1987), pp. 167-185, Academic Press, Boston, MA, 1988.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1, Teubner, Leipzig, 1909; 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.

Daniel Forgues, Table of n, a(n) for n = 1..40882
Dragos Crisan and Radek Erban, On the counting function of semiprimes, arXiv:2006.16491 [math.NT], 2020.
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.
Eric Weisstein's World of Mathematics, Semiprime

Cf. A000720, A001358, A066265, A064911.

A072000 := proc(n) local sp,t ; sp := 0 ; for t from 1 to n do if numtheory[bigomega](t) = 2 then sp := sp+1 ; fi ; od ; sp ; end proc: # R. J. Mathar, Jun 10 2007

semiPrimePi[n_] := Sum[ PrimePi[n/Prime@i] -i + 1, {i, PrimePi@Sqrt@n}]; Array[semiPrimePi, 78] (* Robert G. Wilson v, Jan 03 2006 *)
(* If version >= 7 *) a[n_] := Select[Range[n], PrimeOmega[#] == 2 &] // Length; Table[a[n], {n, 1, 77}] (* Jean-François Alcover, Jun 29 2013 *)
Accumulate[Table[If[PrimeOmega[n]==2,1,0],{n,100}]] (* Harvey P. Dale, Jun 14 2014 *)

for(n=1,100,print1(sum(i=1,n,if(bigomega(i)-2,0,1)),","))

a(n)=my(s=0,i=0); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2 \\ Charles R Greathouse IV, Apr 21 2011

from math import isqrt
from sympy import primepi, prime
def A072000(n): return int(sum(primepi(n//prime(k))-k+1 for k in range(1,primepi(isqrt(n))+1))) # Chai Wah Wu, Jul 23 2024

Let PrimePi(x) denote the number of primes <= x (cf. A000720). Then 2*a(n) = Sum_{ primes p <= n/2 } PrimePi(n/p) + PrimePi(sqrt(n)). [Landau, p. 211]
Let PrimePi(x) denote the number of primes <= x (cf. A000720). Then a(n) = Sum_{i=1..PrimePi(sqrt(n))} (PrimePi(n/prime(i)) - i + 1). - Robert G. Wilson v, Feb 07 2006
a(n) = card { x <= n : bigomega(x) = 2 }.
Asymptotically a(n) ~ n*log(log(n))/log(n). [Landau, p. 211]
Let A be a positive integer. Then card { x <= n : bigomega(x) = A } ~ (n/log(n))*log(log(n))^(A-1)/(A-1)! [Landau, p. 211]
a(n) = A072613(n) + A056811(n). - R. J. Mathar, Jun 10 2007
a(n) = Sum_{i=1..n} A064911(i). - Jonathan Vos Post, Dec 30 2007
a(n)*A064911(n) = A174956(n). - Reinhard Zumkeller, Apr 03 2010

Edited by Robert G. Wilson v, Feb 15 2006

A036352 Number of numbers up to 10^n that are products of two primes.

Original entry on oeis.org

4, 34, 299, 2625, 23378, 210035, 1904324, 17427258, 160788536, 1493776443, 13959990342, 131126017178, 1237088048653, 11715902308080, 111329817298881, 1061057292827269, 10139482913717352, 97123037685177087, 932300026230174178, 8966605849641219022, 86389956293761485464, 833671466551239927908, 8056846659984852885191

Offset: 1

Shyam Sunder Gupta

nonn

Dragos Krisan and Radek Erban, On the counting function of semiprimes, arXiv:2006.16491 [math.NT], 8 Jul 2020.

Essentially the same as A066265.

SemiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}]; Array[ SemiPrimePi[10^#] &, 14] (* Robert G. Wilson v, Feb 12 2015 *)

a(n)=my(s);forprime(p=2,sqrt(10^n),s+=primepi(10^n\p)); s-binomial(primepi(sqrt(10^n)),2) \\ Charles R Greathouse IV, Apr 23 2012

from math import isqrt
from sympy import primepi, primerange
def A036352(n): return int((-(t:=primepi(s:=isqrt(m:=10**n)))*(t-1)>>1)+sum(primepi(m//k) for k in primerange(1, s+1))) # Chai Wah Wu, Aug 16 2024

a(14) from Robert G. Wilson v, May 16 2005
a(15)-a(16) from Donovan Johnson, Mar 18 2010
a(17)-a(18) from A066265, added by Jens Kruse Andersen, Aug 16 2014
a(19)-a(21) from Henri Lifchitz, Jul 04 2015
a(22)-a(23) from Henri Lifchitz, Nov 09 2024

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

A114453 Number of 5-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 4, 76, 963, 11185, 124465, 1349779, 14371023, 150982388, 1570678136, 16218372618, 166497674684, 1701439985694, 17323079621014, 175846040834673, 1780617141307093, 17993699600756449, 181520864946969233

Offset: 0

Robert G. Wilson v, Feb 07 2006

			There are 4 five-almost primes up to 100: 32,48,72 and 80, so a(2) = 4.

Cf. A006880, A066265, A109251, A014614, A116382.

FiveAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k*Prime@l)] - l + 1, {i, PrimePi[n^(1/5)]}, {j, i, PrimePi[(n/Prime@i)^(1/4)]}, {k, j, PrimePi[(n/(Prime@i*Prime@j))^(1/3)]}, {l, k, PrimePi[(n/(Prime@i*Prime@j*Prime@k))^(1/2)]}]; Table[ FiveAlmostPrimePi[10^n], {n, 0, 12}]

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A114453(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,0,1,1,5))) # Chai Wah Wu, Sep 18 2024

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007

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

A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

Original entry on oeis.org

0, 1, 1, 4, 4, 12, 20, 40, 75, 147, 285, 535, 1062, 2006, 3918, 7548, 14595, 28293, 54761, 106452, 206421, 401522, 780966, 1520543, 2962226, 5777162, 11272279, 22009839, 43006972, 84077384, 164482781, 321944211, 630487562, 1235382703

Offset: 0

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

nonn

The partial sum equals the number of Pi_2(2^n) = A125527(n).

			(2^2, 2^3] there is one semiprime, namely 6. 4 was counted in the previous entry.

Dana Jacobsen, Table of n, a(n) for n = 0..62 (first 48 terms from Charles R Greathouse IV, corrected a(47)-a(48))

Cf. A001358, A072000, A066265, A036378, A120033-A120043.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime[i]] - i + 1, {i, PrimePi[Sqrt[n]]}]; t = Table[SemiPrimePi[2^n], {n, 0, 35}]; Rest@t - Most@t

pi2(n)=my(s,i); forprime(p=2, sqrt(n), s+=primepi(n\p); i++); s - i * (i-1)/2
a(n)=pi2(2^(n+1))-pi2(2^n) \\ Charles R Greathouse IV, May 15 2012

use ntheory ":all"; print "$ ",semiprime_count(1+(1<<$), 1<<($+1)),"\n" for 0..48; # _Dana Jacobsen, Mar 04 2019

use ntheory ":all"; my $l=0; for (0..48) { my $c=semiprime_count(1<<($+1)); print "$ ",$c-$l,"\n"; $l=$c; } # Dana Jacobsen, Mar 04 2019

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

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

A120048 Number of 7-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 14, 231, 2973, 35585, 409849, 4600247, 50678212, 550454756, 5913771637, 62981797962, 665997804082, 7001087934965, 73232029374751, 762783057783010, 7916319351632036, 81898808371556517

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 14 seven-almost primes up to 1000: 128, 192, 288, 320, 432, 448, 480, 648, 672, 704, 720, 800, 832 & 972.

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[7, 10^n], {n, 11}]

More terms from Robert G. Wilson v, Jan 07 2007
Example corrected by Harvey P. Dale, Jan 25 2013
a(15)-a(18) from Henri Lifchitz, Mar 18 2025

A120050 Number of 9-almost primes less than or equal to 10^n.

Original entry on oeis.org

0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156, 148930536, 1636170477, 17787688377, 191742524399, 2052389350029, 21838745177567, 231206458686127, 2437121982958248, 25591920108631224, 267840642082525459

Offset: 0

Robert G. Wilson v, Feb 07 2006

nonn

			There are 2 nine-almost primes up to 1000: 512 & 768.

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[9, 10^n], {n, 12}]

a(13) and a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 18 2025

cp's OEIS Frontend

A001358 Semiprimes (or biprimes): products of two primes.

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A072000 Number of semiprimes (A001358) <= n.

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A036352 Number of numbers up to 10^n that are products of two primes.

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

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A114453 Number of 5-almost primes less than or equal to 10^n.

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A120033 Number of semiprimes s such that 2^n < s <= 2^(n+1).

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