A046314 -id:A046314 - cp's OEIS frontend

A069274 13-almost primes (generalization of semiprimes).

8192, 12288, 18432, 20480, 27648, 28672, 30720, 41472, 43008, 45056, 46080, 51200, 53248, 62208, 64512, 67584, 69120, 69632, 71680, 76800, 77824, 79872, 93312, 94208, 96768, 100352, 101376, 103680, 104448, 107520, 112640, 115200

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 13 not necessarily distinct primes.

Divisible by exactly 13 prime powers (not including 1).

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

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), this sequence (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[30000], Plus @@ Last /@ FactorInteger[ # ] == 13 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[116000],PrimeOmega[#]==13&] (* Harvey P. Dale, Mar 11 2019 *)

k=13; start=2^k; finish=130000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A067274(n):
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: 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(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x, 0, 1, 1, 13)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 13.

A069278 17-almost primes (generalization of semiprimes).

Original entry on oeis.org

131072, 196608, 294912, 327680, 442368, 458752, 491520, 663552, 688128, 720896, 737280, 819200, 851968, 995328, 1032192, 1081344, 1105920, 1114112, 1146880, 1228800, 1245184, 1277952, 1492992, 1507328, 1548288, 1605632, 1622016

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 17 not necessarily distinct primes.
Divisible by exactly 17 prime powers (not including 1).
For n = 1..2628 a(n)=2*A069277(n). - Zak Seidov, Jun 25 2017

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

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), this sequence (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[2*10^6],PrimeOmega[#]==17&] (* Harvey P. Dale, Sep 28 2016 *)

k=17; start=2^k; finish=2000000; v=[]
for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069278(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+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,17)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

Product p_i^e_i with Sum e_i = 17.

A069280 19-almost primes (generalization of semiprimes).

Original entry on oeis.org

524288, 786432, 1179648, 1310720, 1769472, 1835008, 1966080, 2654208, 2752512, 2883584, 2949120, 3276800, 3407872, 3981312, 4128768, 4325376, 4423680, 4456448, 4587520, 4915200, 4980736, 5111808, 5971968, 6029312, 6193152

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 19 not necessarily distinct primes.

Divisible by exactly 19 prime powers (not including 1).

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

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), this sequence (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

k=19; start=2^k; finish=8000000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A069280(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,0,1,1,19)))
    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 = 19.

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

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

A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 103, 233, 487, 1072, 2246, 4803, 10202, 21440, 45115, 94434, 197891, 412010, 858846, 1783610, 3700698, 7665755, 15853990, 32750248, 67564405, 139238488, 286625278, 589472979, 1211146741, 2486322304

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Mar 21 2006

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..47

Cf. A046314, A036378, A120033, A120034, A120035, A120036, A120037, A120038, A120039, A120040, A120041, A120042, A120043.

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 *)
t = Table[AlmostPrimePi[10, 2^n], {n, 0, 39}]; Rest@t - Most@t

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120041(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 almostprimepi(n,k): 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(m:=1<Chai Wah Wu, Aug 31 2024

a(n) ~ 2^n log^9 n/(725760 n log 2). [Charles R Greathouse IV, Dec 28 2011]

A281222 Products of 10 distinct primes (squarefree 10-almost primes).

Original entry on oeis.org

6469693230, 6915878970, 8254436190, 8720021310, 9146807670, 9592993410, 10407767370, 10485364890, 10555815270, 11125544430, 11532931410, 11797675890, 11823922110, 12095513430, 12328305990, 12598876290, 12929686770, 13162479330, 13220677470, 13467764310

Offset: 1

Rick L. Shepherd, Jan 17 2017

nonn

			a(1) = 2*3*5*7*11*13*17*19*23*29 = 6469693230 = prime(10)# = A002110(10), the 10th primorial number.

Rick L. Shepherd and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 231 terms from Rick L. Shepherd)
Chai Wah Wu, Algorithms for complementary sequences, arXiv:2409.05844 [math.NT], 2024.

Intersection of A005117 and A046314.

f[om_, lm_ : 0] := Block[{i, j, k, nn, w},
  i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]];
  w = ConstantArray[1, i];
  Union@ Reap[ Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@ w]]; k <= nn,
      Sow[k]; j = 1; w[[-j]]++];
    If[j == i, Break[],
      j++; w[[-j]]++; w = PadRight[w[[;; -j]], i, 1] ],
    {n, Infinity}] ][[-1, 1]] ];
f[10, 10^11] (* Michael De Vlieger, Apr 19 2025 *)

IsInA281222(n) = n > 0 && issquarefree(n) && bigomega(n) == 10

list(lim,pr=10,maxp=lim\vecprod(primes(pr-1)))=if(pr==1, return(primes([2,min(lim\1,maxp)]))); my(v=List(), pr1=pr-1, mx=prod(i=1, pr1, prime(i))); forprime(p=prime(pr), min(lim\mx,maxp), my(u=list(lim\p, pr1, p-1)); for(i=1, #u, listput(v, p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023; corrected by David A. Corneth, Jul 22 2025

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A281222(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) 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,0,1,1,10)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 29 2024

A125149 a(n) is the least k such that the n-almost prime count is positive and equal to the (n-1)-almost prime count. a(0) = 1.

Original entry on oeis.org

1, 2, 10, 15495, 151165506066

Offset: 0

Jonathan Vos Post and Robert G. Wilson v, Jan 07 2007

Unlike any of the prime number races in which any particular form may lead or trail, this sequence demonstrates that although the count of numbers having k prime factors begins by trailing the count for k-1 prime factors, eventually they exchange positions in the race. This can be seen by looking at A126279 or A126280.
The fundamental theorem of arithmetic, or unique factorization theorem, states that every natural number greater than 1 either is itself a prime number, or can be written as a unique product of prime numbers. It had a proof sketched by Euclid, then corrected and completed in "Disquisitiones Arithmeticae" [Carl Friedrich Gauss, 1801]. It fails in many rings of algebraic integers [Ernst Kummer, 1843], a discovery initiating algebraic number theory. Counting the elements in the unique product of prime numbers classifies natural numbers into primes, semiprimes, 3-almost primes and so on. This sequence quantifies a previously undescribed structure to that classification.
We took the first k where the two relevant counts are the same. If instead we took the least k such that the n-almost prime count from k onwards exceeds the (n-1)-almost prime count, the sequence would begin: 3, 34, 15530, ... [see A180126].
The prime count and the semiprime count are identical for 1, 10, 15, 16, 22, 25, 29, 30, 33.
The semiprime count and the 3-almost prime count are identical for 1, 2, 3, 15495, 15496, 15497, 15498, 15508, 15524, 15525, 15529.
The numbers of 3-almost primes and 4-almost primes are equal at 151165506066 and 731 larger numbers, the last one being 151165607041. See A180126. - T. D. Noe, Aug 11 2010
Landau's asymptotic formula suggests that a(n) is about exp(exp(n-1)). - Charles R Greathouse IV, Mar 14 2011

			a(1) = 2 since 1 has no prime factors and 2 has one prime factor, therefore prime factor counts of 0 and 1 occur equally often in the first 2 integers.
a(2) = 10 since there are 4 primes {2, 3, 5 & 7} and 4 semiprimes {4, 6, 9 & 10} less than or equal to 10.
a(4) = 151165506066 since there are 32437255807 4-almost primes and 3-almost primes <= a(4).

Andrew Granville and Greg Martin, Prime Number Races, arXiv:math/0408319 [math.NT], Aug 24 2004.
Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic.
Eric Weisstein's World of Mathematics, Modular Prime Counting Function.
Eric Weisstein's World of Mathematics, Prime Factor.

Cf. A126279, A126280, A117526.
Sequences listing r-almost primes, that is, k such that A001222(k) = 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), A069281 (r = 20).
Cf. A180126.

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 *)
f[n_] := Block[{k = 2^n}, While[AlmostPrimePi[n, k] < AlmostPrimePi[n - 1, k], k++ ]; k];

Changed 33 to 34 in a comment. - T. D. Noe, Aug 11 2010

Edited by Peter Munn, Dec 17 2022

A046336 Palindromes with exactly 10 prime factors (counted with multiplicity).

Original entry on oeis.org

8448, 40704, 42624, 46464, 63936, 65856, 69696, 234432, 255552, 297792, 426624, 639936, 2152512, 2300032, 2327232, 2346432, 2570752, 2726272, 2741472, 2783872, 2939392, 2996992, 4072704, 4209024, 4266624, 4811184, 4897984, 6129216, 6167616, 6186816, 6334336

Offset: 1

Patrick De Geest, Jun 15 1998

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A046314, A046323.

Select[Range[2, 10000000], IntegerDigits[ # ] == Reverse[IntegerDigits[ # ]] && Plus @@ Transpose[FactorInteger[ # ]][[2]] == 10 &] (* Tanya Khovanova, Sep 07 2007 *)

A101745 Indices of triangular numbers which are 10-almost primes. Indices of A101744.

Original entry on oeis.org

255, 384, 511, 575, 639, 728, 767, 896, 1088, 1295, 1376, 1407, 1599, 1700, 1727, 1792, 1919, 1920, 2015, 2024, 2375, 2431, 2672, 2815, 2880, 2915, 2944, 2975, 3104, 3159, 3199, 3327, 3375, 3392, 3456, 3583, 3744, 3999, 4031, 4032, 4160, 4223, 4256

Offset: 1

Jonathan Vos Post, Dec 14 2004

			a(1) = 255 because that is the smallest index of a triangular number which is also a 10-almost prime; specifically T(255) = 255*(255+1)/2 = 32640 = 2^7 * 3 * 5 * 17.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Cf. A101744, A000217, A046314, A076550, A069904.

F:=List([1..4300],n->Length(Factors(n*(n+1)/2)));; a:=Filtered([1..Length(F)],i->F[i]=10); # Muniru A Asiru, Dec 22 2018

[n: n in [2..4500] | &+[d[2]: d in Factorization((n*(n+1)))] eq 11]; // Vincenzo Librandi, Dec 22 2018

BigOmega[n_Integer]:=Plus@@Last[Transpose[FactorInteger[n]]]; Do[ t=n*(n+1)/2; If[BigOmega[t]==10, Print[n, " ", t];];, {n, 2, 5000}]; (* Ray Chandler, Dec 14 2004 *)
Flatten[Position[Accumulate[Range[5000]],?(PrimeOmega[#]==10&)]] (* _Harvey P. Dale, May 12 2011 *)

a(n)*(a(n)+1)/2 has exactly 10 prime factors.

{ m : A069904(m) = 10 }. - Alois P. Heinz, Aug 05 2019

More terms from Ray Chandler, Dec 14 2004

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A069274 13-almost primes (generalization of semiprimes).

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A069278 17-almost primes (generalization of semiprimes).

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A069280 19-almost primes (generalization of semiprimes).

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

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A091538 Triangle built from m-primes as columns.

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A120041 Number of 10-almost primes k such that 2^n < k <= 2^(n+1).

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A281222 Products of 10 distinct primes (squarefree 10-almost primes).

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