A046310 -id:A046310 - 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

A110296 8-almost primes pqrstuv*w relatively prime to p+q+r+s+t+u+v+w.

Original entry on oeis.org

384, 640, 864, 1408, 1664, 2016, 2176, 2400, 2432, 2944, 3240, 3712, 3744, 3968, 4374, 4536, 4736, 5248, 5280, 5472, 5504, 5600, 6016, 6240, 6784, 7128, 7392, 7552, 7808, 7840, 8424, 8576, 8800, 8928, 9088, 9120, 9344, 10112, 10400, 10584, 10624

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v, w are not necessarily distinct. The 8-almost primes are A046310. The converse, A110297, is 8-almost primes p*q*r*s*t*u*v*w which are not relatively prime to p+q+r+s+t+u+v+w.

			864 is an element of this sequence because 864 = 2^5 * 3^3, so the sum of prime factors is 2 + 2 + 2 + 2 + 2 + 3 + 3 + 3 = 19 which is prime, hence relatively prime to 864. That is the same sum of prime factors as 640 = 2^7 * 5, hence 640 is also a member of this sequence. The sum of prime factors need not be prime for this membership, for example, 2432 = 2^7 * 19 has sum of prime factors 2 + 2 + 2 + 2 + 2 + 2 + 2 + 19 = 33 = 3 * 11, which is composite, yet relatively prime to 2432.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046310, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\128, forprime(q=2, min(p, lim\64\p), my(pq=p*q); forprime(r=2, min(lim\pq\32, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\16, r), my(pqrs=pqr*s); forprime(t=2, min(lim\pqrs\8, s), my(pqrst=pqrs*t); forprime(u=2, min(lim\pqrst\4, t), my(pqrstu=pqrst*u); forprime(w=2,min(lim\pqrstu\2,u), my(pqrstuw=pqrstu*w,n); forprime(x=2,min(lim\pqrstuw,w), n=pqrstuw*x; if(gcd(n, p+q+r+s+t+u+w+x)==1, listput(v, n)))))))))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Corrected and extended by Ray Chandler, Jul 20 2005

A110297 8-almost primes pqrstuv*w not relatively prime to p+q+r+s+t+u+v+w.

Original entry on oeis.org

256, 576, 896, 960, 1296, 1344, 1440, 1600, 1944, 2112, 2160, 2240, 2496, 2916, 3024, 3136, 3168, 3264, 3360, 3520, 3600, 3648, 4000, 4160, 4416, 4704, 4752, 4860, 4896, 4928, 5040, 5400, 5440, 5568, 5616, 5824, 5952, 6000, 6080, 6561, 6624, 6804, 7056

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v, w are not necessarily distinct. The 8-almost primes are A046310. The converse, A110296, is 8-almost primes p*q*r*s*t*u*v*w which are relatively prime to p+q+r+s+t+u+v+w.

			576 = 2^6 * 3^2 is an element of this sequence because its sum of prime factors is 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 = 18 = 2 * 3^2 which is a factor of 576 and not relatively prime to 576.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A046310, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110290, A110296.

list(lim)=my(v=List()); forprime(p=2, lim\128, forprime(q=2, min(p, lim\64\p), my(pq=p*q); forprime(r=2, min(lim\pq\32, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\16, r), my(pqrs=pqr*s); forprime(t=2, min(lim\pqrs\8, s), my(pqrst=pqrs*t); forprime(u=2, min(lim\pqrst\4, t), my(pqrstu=pqrst*u); forprime(w=2,min(lim\pqrstu\2,u), my(pqrstuw=pqrstu*w,n); forprime(x=2,min(lim\pqrstuw,w), n=pqrstuw*x; if(gcd(n, p+q+r+s+t+u+w+x)>1, listput(v, n)))))))))); Set(v) \\ Charles R Greathouse IV, Feb 01 2017

Extended by Ray Chandler, Jul 20 2005

A123322 Products of 8 distinct primes (squarefree 8-almost primes).

Original entry on oeis.org

9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 18888870, 20030010, 20281170, 20930910, 21111090, 21411390, 21637770, 21951930, 23130030, 23393370, 23993970, 24534510, 25555530, 25571910

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046310.

			a(1) = 9699690 = 2*3*5*7*11*13*17*19 = A002110(8).

Robert G. Wilson v, Table of n, a(n) for n = 1..29422 (first 10000 terms from Rick Shepherd)
Index to sequences related to prime signature

Cf. A001221, A001222, A005117, A046310, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123321 (k=7), A115343 (k=9).

N:= 3*10^7: # to get all terms  <= N
pmax:= floor(N/mul(ithprime(i),i=1..7)):
Primes:= select(isprime,[2,seq(i,i=3..pmax,2)]):
sort(select(`<`,map(convert,combinat:-choose(Primes,8),`*`),N)); # Robert Israel, Dec 18 2018

f8Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f8Q[n], AppendTo[lst, n]], {n, 10!, 11!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Take[ Sort[ Times @@@ Subsets[ Prime@ Range@ 15, {8}]], 22] (* Robert G. Wilson v, Dec 18 2018 *)

is(n)=issquarefree(n)&&omega(n)==8 \\ Charles R Greathouse IV, Feb 01 2017, corrected (following an observation from Zak Seidov) by M. F. Hasler, Dec 19 2018

is(n) = my(f = factor(n)); omega(f) == 8 && bigomega(f) == 8 \\ David A. Corneth, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123322(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,0,1,1,8)))
    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

Edited by Robert Israel, Dec 18 2018

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

A120039 Number of 8-almost primes 8ap such that 2^n < 8ap <= 2^(n+1).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 8, 22, 47, 101, 229, 473, 1044, 2171, 4634, 9796, 20513, 43020, 89684, 187361, 388633, 807508, 1671160, 3455934, 7135226, 14708436, 30286472, 62280024, 127944070, 262543635, 538266791, 1102507513, 2256357137

Offset: 0

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

nonn

The partial sum equals the number of Pi_8(2^n).

			(2^8, 2^9] there is one semiprime, namely 384. 256 was counted in the previous entry.

Cf. A046310, 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[8, 2^n], {n, 0, 30}]; Rest@t - Most@t

A113740 Pierpont 8-almost primes. 8-almost primes of form (2^K)*(3^L)+1.

Original entry on oeis.org

1999004627104432129, 4052555153018976268, 8754997675608244225, 9606056659007943745, 11832592569282330625, 22769912080611422209, 68309736241834266625, 354577405862133891073, 12449449430074295092225

Offset: 1

Jonathan Vos Post, Nov 08 2005

nonn

			a(1) = 1999004627104432129 = (2^18)*(3^27)+1 = 7 * 13 * 19 * 109 * 127 * 181 * 6949 * 66403.
a(2) = 4052555153018976268 = (2^0)*(3^39)+1 = 2 * 2 * 7 * 79 * 157 * 2887 * 10141 * 398581.
a(3) = 8754997675608244225 = (2^55)*(3^5)+1 = 5 * 5 * 11 * 11 * 1201 * 1229 * 16451 * 119191.
a(4) = 9606056659007943745 = (2^6)*(3^36)+1 = 5 * 13 * 17 * 89 * 109 * 281 * 18793 * 169693.
a(13) = 717897987691852588770250 = (2^0)*(3^50)+1 = 2 * 5 * 5 * 5 * 101 * 1181 * 394201 * 61070817601.
a(29) = 1570042899082081611640534564 = (2^0)*(3^57)+1 = 2 * 2 * 7 * 2851 * 3079 * 53923 * 101917 * 1162320517.

Charles R Greathouse IV, Table of n, a(n) for n = 1..993
Eric Weisstein's World of Mathematics, Pierpont Prime
Eric Weisstein's World of Mathematics, Almost Prime

Intersection of A046310 and A055600.
A005109 gives the Pierpont primes, which are primes of the form (2^K)*(3^L)+1.
A113432 gives the Pierpont semiprimes, 2-almost primes of the form (2^K)*(3^L)+1.
A112797 gives the Pierpont 3-almost primes, of the form (2^K)*(3^L)+1.
A111344 gives the Pierpont 4-almost primes, of the form (2^K)*(3^L)+1.
A111345 gives the Pierpont 5-almost primes, of the form (2^K)*(3^L)+1.
A111346 gives the Pierpont 6-almost primes, of the form (2^K)*(3^L)+1.
A113739 gives the Pierpont 7-almost primes, of the form (2^K)*(3^L)+1.
A113741 gives the Pierpont 9-almost primes, of the form (2^K)*(3^L)+1.

list(lim)=my(v=List(), L=lim\1-1); for(e=0, logint(L, 3), my(t=3^e); while(t<=L, if(bigomega(t+1)==8, listput(v, t+1)); t*=2)); Set(v) \\ Charles R Greathouse IV, Feb 06 2017

a(n) is in this sequence iff there exist nonnegative integers K and L such that Omega((2^K)*(3^L)+1) = 8.

Extended by Ray Chandler, Nov 08 2005

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