A014612 -id:A014612 - cp's OEIS frontend

A069272 11-almost primes (generalization of semiprimes).

2048, 3072, 4608, 5120, 6912, 7168, 7680, 10368, 10752, 11264, 11520, 12800, 13312, 15552, 16128, 16896, 17280, 17408, 17920, 19200, 19456, 19968, 23328, 23552, 24192, 25088, 25344, 25920, 26112, 26880, 28160, 28800, 29184, 29696

Offset: 1

Rick L. Shepherd, Mar 12 2002

nonn

Product of 11 not necessarily distinct primes.

Divisible by exactly 11 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), this sequence (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). - Jason Kimberley, Oct 02 2011

Select[Range[9000], Plus @@ Last /@ FactorInteger[ # ] == 11 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[30000],PrimeOmega[#]==11&] (* Harvey P. Dale, Jul 13 2013 *)

k=11; start=2^k; finish=30000; v=[]; for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v

is(n)=bigomega(n)==11 \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A069272(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, 11)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 11.

a(n) ~ 3628800n log n / (log log n)^10. - Charles R Greathouse IV, May 06 2013

A337461 Number of pairwise coprime ordered triples of positive integers summing to n.

Original entry on oeis.org

0, 0, 0, 1, 3, 3, 9, 3, 15, 9, 21, 9, 39, 9, 45, 21, 45, 21, 87, 21, 93, 39, 87, 39, 153, 39, 135, 63, 153, 57, 255, 51, 207, 93, 225, 93, 321, 81, 291, 135, 321, 105, 471, 105, 393, 183, 381, 147, 597, 147, 531, 213, 507, 183, 759, 207, 621, 273, 621, 231

Offset: 0

Gus Wiseman, Sep 02 2020

nonn

			The a(3) = 1 through a(9) = 9 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)  (1,1,7)
           (1,2,1)  (1,3,1)  (1,2,3)  (1,5,1)  (1,2,5)  (1,3,5)
           (2,1,1)  (3,1,1)  (1,3,2)  (5,1,1)  (1,3,4)  (1,5,3)
                             (1,4,1)           (1,4,3)  (1,7,1)
                             (2,1,3)           (1,5,2)  (3,1,5)
                             (2,3,1)           (1,6,1)  (3,5,1)
                             (3,1,2)           (2,1,5)  (5,1,3)
                             (3,2,1)           (2,5,1)  (5,3,1)
                             (4,1,1)           (3,1,4)  (7,1,1)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

A000212 counts the unimodal instead of coprime version.
A220377*6 is the strict case.
A307719 is the unordered version.
A337462 counts these compositions of any length.
A337563 counts the case of partitions with no 1's.
A337603 only requires the *distinct* parts to be pairwise coprime.
A337604 is the intersecting instead of coprime version.
A014612 ranks 3-part partitions.
A302696 ranks pairwise coprime partitions.
A327516 counts pairwise coprime partitions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A000217, A001399, A001840, A014311, A101268, A284825, A337562, A326675, A333227, A337601, A337602.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,30}]

A307719 Number of partitions of n into 3 mutually coprime parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 1, 3, 2, 4, 2, 7, 2, 8, 4, 8, 4, 15, 4, 16, 7, 15, 7, 26, 7, 23, 11, 26, 10, 43, 9, 35, 16, 38, 16, 54, 14, 49, 23, 54, 18, 79, 18, 66, 31, 64, 25, 100, 25, 89, 36, 85, 31, 127, 35, 104, 46, 104, 39, 167, 36, 125, 58, 129, 52, 185, 45

Offset: 0

Wesley Ivan Hurt, Apr 24 2019

The Heinz numbers of these partitions are the intersection of A014612 (triples) and A302696 (pairwise coprime). - Gus Wiseman, Oct 16 2020

			There are 2 partitions of 9 into 3 mutually coprime parts: 7+1+1 = 5+3+1, so a(9) = 2.
There are 4 partitions of 10 into 3 mutually coprime parts: 8+1+1 = 7+2+1 = 5+4+1 = 5+3+2, so a(10) = 4.
There are 2 partitions of 11 into 3 mutually coprime parts: 9+1+1 = 7+3+1, so a(11) = 2.
There are 7 partitions of 12 into 3 mutually coprime parts: 10+1+1 = 9+2+1 = 8+3+1 = 7+4+1 = 6+5+1 = 7+3+2 = 5+4+3, so a(12) = 7.

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)
Index entries for sequences related to partitions

A023022 is the version for pairs.
A220377 is the strict case, with ordered version A220377*6.
A327516 counts these partitions of any length, with strict version A305713 and Heinz numbers A302696.
A337461 is the ordered version.
A337563 is the case with no 1's.
A337599 is the pairwise non-coprime instead of pairwise coprime version.
A337601 only requires the distinct parts to be pairwise coprime.
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A002865 counts partitions with no 1's, with strict case A025147.
A007359 and A337485 count pairwise coprime partitions with no 1's.
A200976 and A328673 count pairwise non-coprime partitions.
Cf. A007304, A014612, A078374, A082024, A284825, A304709, A337462, A337605.

N:= 200: # to get a(0)..a(N)
A:= Array(0..N):
for a from 1 to N/3 do
  for b from a to (N-a)/2 do
    if igcd(a,b) > 1 then next fi;
    ab:= a*b;
    for c from b to N-a-b do
       if igcd(ab,c)=1 then A[a+b+c]:= A[a+b+c]+1 fi
od od od:
convert(A,list); # Robert Israel, May 09 2019

Table[Sum[Sum[Floor[1/(GCD[i, j] GCD[j, n - i - j] GCD[i, n - i - j])], {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 100}]
Table[Length[Select[IntegerPartitions[n,{3}],CoprimeQ@@#&]],{n,0,100}] (* Gus Wiseman, Oct 15 2020 *)

a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} [gcd(i,j) * gcd(j,n-i-j) * gcd(i,n-i-j) = 1], where [] is the Iverson bracket.

a(n > 2) = A220377(n) + 1. - Gus Wiseman, Oct 15 2020

A069273 12-almost primes (generalization of semiprimes).

Original entry on oeis.org

4096, 6144, 9216, 10240, 13824, 14336, 15360, 20736, 21504, 22528, 23040, 25600, 26624, 31104, 32256, 33792, 34560, 34816, 35840, 38400, 38912, 39936, 46656, 47104, 48384, 50176, 50688, 51840, 52224, 53760, 56320, 57600, 58368, 59392

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 12 not necessarily distinct primes.
Divisible by exactly 12 prime powers (not including 1).
Any 12-almost prime can be represented in at least one way as a product of two 6-almost primes A046306, three 4-almost primes A014613, four 3-almost primes A014612, or six semiprimes A001358. - Jonathan Vos Post, Dec 11 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), this sequence (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). - Jason Kimberley, Oct 02 2011

Select[Range[20000], Plus @@ Last /@ FactorInteger[ # ] == 12 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[60000],PrimeOmega[#]==12&] (* Harvey P. Dale, May 01 2019 *)

k=12; start=2^k; finish=70000; 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 A069273(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, 12)))
    return bisection(f, n, n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 12.

A069279 Products of exactly 18 primes (generalization of semiprimes).

Original entry on oeis.org

262144, 393216, 589824, 655360, 884736, 917504, 983040, 1327104, 1376256, 1441792, 1474560, 1638400, 1703936, 1990656, 2064384, 2162688, 2211840, 2228224, 2293760, 2457600, 2490368, 2555904, 2985984, 3014656, 3096576, 3211264, 3244032, 3317760, 3342336, 3440640

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 18 not necessarily distinct primes.

Divisible by exactly 18 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.

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

Select[Range[31*10^5],PrimeOmega[#]==18&] (* Harvey P. Dale, Apr 05 2015 *)

k=18; start=2^k; finish=4000000; 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 A069279(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,0,1,1,18)))
    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 = 18.

A069281 20-almost primes (generalization of semiprimes).

Original entry on oeis.org

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

A069275 14-almost primes (generalization of semiprimes).

Original entry on oeis.org

16384, 24576, 36864, 40960, 55296, 57344, 61440, 82944, 86016, 90112, 92160, 102400, 106496, 124416, 129024, 135168, 138240, 139264, 143360, 153600, 155648, 159744, 186624, 188416, 193536, 200704, 202752, 207360, 208896, 215040, 225280

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 14 not necessarily distinct primes.
Divisible by exactly 14 prime powers (not including 1).
Any 14-almost prime can be represented in several ways as a product of two 7-almost primes A046308; and in several ways as a product of seven semiprimes A001358. - Jonathan Vos Post, Dec 11 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), this sequence(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[50000], Plus @@ Last /@ FactorInteger[ # ] == 14 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)

k=14; start=2^k; finish=240000; 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 A069275(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,0,1,1,14)))
    return bisection(f,n,n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 14.

A069276 15-almost primes (generalization of semiprimes).

Original entry on oeis.org

32768, 49152, 73728, 81920, 110592, 114688, 122880, 165888, 172032, 180224, 184320, 204800, 212992, 248832, 258048, 270336, 276480, 278528, 286720, 307200, 311296, 319488, 373248, 376832, 387072, 401408, 405504, 414720, 417792, 430080

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 15 not necessarily distinct primes.
Divisible by exactly 15 prime powers (not including 1).
Any 15-almost prime can be represented in several ways as a product of three 5-almost primes A014614, and in several ways as a product of five 3-almost primes A014612. - Jonathan Vos Post, Dec 11 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), this sequence (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[90000], Plus @@ Last /@ FactorInteger[ # ] == 15 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[450000],PrimeOmega[#]==15&] (* Harvey P. Dale, Aug 14 2019 *)

k=15; start=2^k; finish=500000; 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 A069276(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,0,1,1,15)))
    return bisection(f,n,n) # Chai Wah Wu, Nov 03 2024

Product p_i^e_i with Sum e_i = 15.

A069277 16-almost primes (generalization of semiprimes).

Original entry on oeis.org

65536, 98304, 147456, 163840, 221184, 229376, 245760, 331776, 344064, 360448, 368640, 409600, 425984, 497664, 516096, 540672, 552960, 557056, 573440, 614400, 622592, 638976, 746496, 753664, 774144, 802816, 811008, 829440, 835584, 860160

Offset: 1

Rick L. Shepherd, Mar 13 2002

nonn

Product of 16 not necessarily distinct primes.
Divisible by exactly 16 prime powers (not including 1).
Any 16-almost prime can be represented in several ways as a product of two 8-almost primes A046310; in several ways as a product of four 4-almost primes A014613; and in several ways as a product of eight 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. A014610, 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), this sequence (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[300000], Plus @@ Last /@ FactorInteger[ # ] == 16 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[10^6],PrimeOmega[#]==16&] (* Harvey P. Dale, Jan 30 2015 *)

k=16; start=2^k; finish=1000000; 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 A069277(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,0,1,1,16)))
    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 = 16.

A069274 13-almost primes (generalization of semiprimes).

Original entry on oeis.org

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.

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