A069272 -id:A069272 - cp's OEIS frontend

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

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

A120042 Number of 11-almost primes 11ap such that 2^n < 11ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 103, 234, 490, 1078, 2261, 4844, 10294, 21659, 45609, 95580, 200422, 417715, 871452, 1811412, 3761623, 7798409, 16142081, 33373093, 68906782, 142120436, 292797806, 602653984, 1239225631

Offset: 0

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

nonn

The partial sum equals the number of Pi_11(2^n).

			(2^11, 2^12] there is one semiprime, namely 3072. 2048 was counted in the previous entry.

Chai Wah Wu, Table of n, a(n) for n = 0..50

Cf. A069272, 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[11, 2^n], {n, 0, 30}]; Rest@t - Most@t

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120042(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, Jun 17 2025

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

A349030 Lucas-Carmichael numbers with 11 prime factors.

Original entry on oeis.org

20576473996736735, 42380075646230399, 75943207554554879, 83668951228080959, 96195222056687039, 116436396482735615, 132525862783734959, 134052021887096159, 162544912900261199, 175900784368936319, 186326804496197519, 190523141606006495, 196467189590024639

Offset: 1

Tim Johannes Ohrtmann, Nov 06 2021

nonn

			20576473996736735 = 5*7*11*17*23*31*47*53*71*107*233 and 6, 8, 12, 18, 24, 32, 48, 54, 72, 108, and 234 all divide 20576473996736736.

Daniel Suteu, Table of n, a(n) for n = 1..2124 (all terms < 2^64).

Intersection of A006972 and A069272.
Cf. A216928 (least Lucas-Carmichael number with n prime factors).
Cf. A216925, A216926, A216927, A217002, A217003, A217091, A349028, A349029 (Lucas-Carmichael numbers with 3-10 prime factors).

PARI
```
is(n)={omega(n)==11&&is_A006972(n)}
```

A101696 a(n) = sum(i=1,n)(i-th i-almost prime). Cumulative sums of A101695.

Original entry on oeis.org

2, 8, 26, 66, 174, 398, 878, 2174, 4862, 10494, 22014, 45054, 98302, 222718, 480766, 1021438, 2127358, 4355582, 8943102, 18773502, 38696446, 79590910, 175142398, 368080382, 764442110, 1586525694, 3247470078, 6644856318, 13489960446

Offset: 1

Jonathan Vos Post, Dec 12 2004, Sep 28 2006

nonn

It seems that this sum can never be a prime after a(1) = 2, since the n-th n-almost prime is always even. The number of prime factors (with multiplicity) of a(n) is 1, 3, 2, 3, 3, 2, 2, 2, 4, 5, 4, 4, 3, 3, 5, 4, 3, 4, 7, 4, 2, 5, 5, 2, 3, 7, 4, 3, 4.

This is the diagonalization of the set of sequences {j-almost prime(k)}. The cumulative sums of this sequence are in A101696. a(1)=2 is prime. a(2)=8 is a 3-almost prime. a(3)=26 is a semiprime. a(4)=66 is a 3-almost prime. a(5)= 174 is a 3-almost prime. a(6)=398 is a semiprime. a(7)=878 is a semiprime. a(8)=2174 is a semiprime. a(9)=4862 is a 4-almost prime. a(10)=10494 is a 5-almost prime. a(11)=22014 is a 4-almost prime. a(12)=45054 is a 3-almost prime. a(13)=98302 is a 3-almost prime. a(14)=222718 is a 3-almost prime. a(15)=480766 is a 5-almost prime. a(16)=1021438 is a 4-almost prime. a(17)=2127358 is a 3-almost prime. a(18)=4355582 is a 4-almost prime. a(19)=8943102 is a 7-almost prime. a(20)=18773502 is a 4-almost prime. 21-almost numbers are not yet listed in the OEIS.

			a(1) = first 1-almost prime = first prime = A000040(1) = 2.
a(2) = 2 + 2nd 2-almost prime = 2 + A001358(2) = 2+ 6 = 8.
a(3) = a(2) + 3rd 3-almost prime = 8+A014612(3) = 8+18 = 26.
a(4) = a(3) + 4th 4-almost prime = 26+A014613(4) = 26+40 = 66.
a(5) = a(4) + 5th 5-almost prime = 66+A014614(5) = 66+108=174.
...
a(12) = a(11) + 12th 12-almost prime = 22014 + 23040 = 45054 (the first nontrivial palindrome in the sequence).

Eric Weisstein's World of Mathematics, Almost Prime.

a(1) = first 1-almost prime = first prime = A000040(1). a(2) = a(1) + 2nd 2-almost prime = a(1) + 2nd semiprime = A000040(1)+A001358(2). a(3) = a(2) + 3rd 3-almost prime = a(2) + A014612(3). a(4) = a(3) + 4th 4-almost prime = a(3) + A014613(4)... a(n) = a(n-1) + n-th n-almost prime.

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A337112 Smallest term of A337081 that has exactly n prime factors, or 0 if no such term exists.

Original entry on oeis.org

0, 4, 0, 90, 675, 1134, 6318, 4374, 32805, 255879, 1003833, 531441, 327544803, 20751953125, 225830078125, 91552734375, 1068115234375, 23651123046875, 316619873046875, 1697540283203125, 13256072998046875, 85353851318359375, 541210174560546875, 4518032073974609375, 58233737945556640625

Offset: 1

Matej Veselovac, Aug 16 2020

nonn

a(n) is the smallest product of n primes that has unordered factorizations whose sums of factors are the same (is a term of A337080) and all of whose proper divisors have the complementary property: that every unordered factorization has a distinct sum of factors (i.e., all proper divisors are terms of A337037).

Cf. A337113 (factors of terms).
Cf. A337037, A337080, A337081.
Cf. A056472 (all factorizations of n).
Cf. r-almost primes: 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).

a(14) onward from David A. Corneth, Aug 26 2020

A338443 Carmichael numbers with 11 prime factors.

Original entry on oeis.org

60977817398996785, 105083995864811041, 107473646345582881, 132819104923908481, 145671955835893201, 161802381510126721, 165167398073764801, 206063729626916161, 263076030916096321, 292433912163313921, 292561243007134465, 337365329710615921, 388219799621120545

Offset: 1

Tim Johannes Ohrtmann, Oct 28 2020

nonn

			60977817398996785 = 5*7*17*19*23*37*53*73*79*89*233 and 4, 6, 16, 18, 22, 36, 52, 72, 78, 88, 232 all divide 60977817398996784.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier; terms 1..1142 from Daniel Suteu)
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Richard Pinch, Tables relating to Carmichael numbers.

Cf. A002997 (Carmichael numbers).
Cf. A006931 (Least Carmichael number with n prime factors).
Cf. A299710 (Number of terms less than 10^n).
Cf. A087788, A074379, A112428, A112429, A112430, A112431, A112432, A338442 (Carmichael numbers with 3-10 prime factors).

PARI
```
is(n)={omega(n)==11&&is_A002997(n)}
```

Equals A002997 intersect A069272.

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cp's OEIS Frontend

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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A120042 Number of 11-almost primes 11ap such that 2^n < 11ap <= 2^(n+1).

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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.

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A349030 Lucas-Carmichael numbers with 11 prime factors.

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