A014614 -id:A014614 - cp's OEIS frontend

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

1, 3, 9, 20, 54, 112, 240, 648, 1344, 2816, 5760, 12800, 26624, 62208, 129024, 270336, 552960, 1114112, 2293760, 4915200, 9961472, 20447232, 47775744, 96468992, 198180864, 411041792, 830472192, 1698693120, 3422552064, 7046430720

Offset: 0

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

nonn

A k-almost prime is a positive integer that has exactly k prime factors counted with multiplicity.

			a(0) = 1 since one is the multiplicative identity,
a(1) = 2nd 1-almost prime is the second prime number = A000040(2) = 3,
a(2) = 3rd 2-almost prime = 3rd semiprime = A001358(3) = 9 = {3*3}.
a(3) = 4th 3-almost prime = A014612(4) = 20 = {2*2*5}.
a(4) = 5th 4-almost prime = A014613(5) = 54 = {2*3*3*3},
a(5) = 6th 5-almost prime = A014614(6) = 112 = {2*2*2*2*7}, ....

Chai Wah Wu, Table of n, a(n) for n = 0..1000 (terms 0..228 from Robert G. Wilson v)
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A078840, A078842, A078843, A078844, A078445, A078846, A101695.

f[n_] := Plus @@ Last /@ FactorInteger@n; t = Table[{}, {40}]; Do[a = f[n]; AppendTo[ t[[a]], n]; t[[a]] = Take[t[[a]], 10], {n, 2, 148*10^8}]; Table[ t[[n, n + 1]], {n, 30}] (* Robert G. Wilson v, Feb 11 2006 *)
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-1, n]], {n, 30}]; lst (* Robert G. Wilson v, Nov 13 2007 *)

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A078841(n):
    if n <= 1: return (n<<1)+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)))
    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,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 as n->inf. of a(n+1)/a(n) = 2. - Robert G. Wilson v, Nov 13 2007

a(14)-a(29) from Robert G. Wilson v, Feb 11 2006

A101638 Number of distinct 4-almost primes A014613 which are factors of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1

Offset: 1

Jonathan Vos Post, Dec 10 2004

This is the inverse Moebius transform of A101637. If we take the prime factorization of n = (p1^e1)*(p2^e2)* ... * (pj^ej) then a(n) = |{k: ek>=4}| + ((j-1)/2)*|{k: ek>=3}| + C(|{k: ek>=2}|,2) + C(j,4). The first term is the number of distinct 4th powers of primes in the factors of n (the first way of finding a 4-almost prime). The second term is the number of distinct cubes of primes, each of which can be multiplied by any of the other distinct primes, halved to avoid double-counts (the second way of finding a 4-almost prime). The third term is the number of distinct pairs of squares of primes in the factors of n (the third way of finding a 4-almost prime). The 4th term is the number of distinct products of 4 distinct primes, which is the number of combinations of j primes in the factors of n taken 4 at a time, A000332(j), (the 4th way of finding a 4-almost prime).

			a(96) = 2 because 96 = 16 * 6 hence divisible by the 4-almost prime 16 and also 96 = 24 * 4 hence divisible by the 4-almost prime 24.

Hardy, G. H. and Wright, E. M. Section 17.10 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. A. Bender and J. R. Goldman, On the Applications of Moebius Inversion in Combinatorial Analysis, Amer. Math. Monthly 82, 789-803, 1975.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Moebius Transform.

Cf. A101638, A014613, A000332, A086971, A100605, A000040, A001358, A014612, A014614.

a(n)=my(f=factor(n)[,2], v=apply(k->sum(i=1,#f,f[i]>k), [0..3])); v[4] + v[3]*(v[1]-1) + binomial(v[2],2) + v[2]*binomial(v[1]-1,2) + binomial(v[1],4) \\ Charles R Greathouse IV, Sep 14 2015

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

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

A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 3, 4, 5, 2, 6, 5, 6, 1, 7, 3, 8, 4, 7, 8, 9, 2, 9, 10, 5, 6, 10, 7, 11, 1, 11, 12, 13, 3, 12, 14, 15, 4, 13, 8, 14, 9, 10, 16, 15, 2, 17, 11, 18, 12, 16, 5, 19, 6, 20, 21, 17, 7, 18, 22, 13, 1, 23, 14, 19, 15, 24, 16, 20, 3, 21, 25, 17, 18, 26, 19, 22, 4, 8, 27, 23

Offset: 1

Naohiro Nomoto, Jan 11 2001

Equivalently, the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - Gus Wiseman, Dec 28 2018

There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - Peter Munn, Aug 04 2019

			3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.
From _Gus Wiseman_, Dec 28 2018: (Start)
Column n lists the a(n) positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity:
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
  ---------------------------------------------------------------------
  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20
        2     3  4  5     6  9   7   8   11  10  14      13  12  17  18
              2     3     4  6   5       7   9   10      11  8   13  12
                    2        4   3       5   6   9       7       11  8
                                 2       3   4   6       5       7
                                         2       4       3       5
                                                         2       3
                                                                 2
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Positions of 1's are A000079.
Cf. A001358, A014612, A014613, A014614.
Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A078843, A209934, A302242, A322838, A322839, A322840.
Equivalent sequence restricted to squarefree numbers: A340313.

p:= proc() 0 end:
a:= proc(n) option remember; local t;
      t:= numtheory[bigomega](n);
      p(t):= p(t)+1
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015

p[] = 0; a[n] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 24 2017, after Alois P. Heinz *)

a(n) = my(k=bigomega(n)); sum(i=1, n, bigomega(i)==k); \\ Michel Marcus, Jun 27 2024

from math import prod, isqrt
from sympy import isprime, primepi, primerange, integer_nthroot, primeomega
def A058933(n):
    if n==1: return 1
    if isprime(n): return primepi(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(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,primeomega(n)))) # Chai Wah Wu, Aug 28 2024

Ordinal transform of A001222 (bigomega). - Franklin T. Adams-Watters, Aug 28 2006

If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - Peter Munn, Aug 05 2019

Name edited by Peter Munn, Dec 30 2022

A074969 Numbers with six distinct prime divisors.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 60060, 62790, 66990, 67830, 71610, 72930, 78540, 79170, 81510, 82110, 84630, 85470, 87780, 90090, 91770, 92820, 94710, 98670, 99330, 101010, 102102, 103530, 103740, 106260, 106590, 108570

Offset: 1

Zak Seidov, Oct 04 2002

nonn

The smallest number with six distinct prime divisors is the product of the first six primes, 2*3*5*7*11 = 30030.

The smallest number with seven distinct prime divisors is the product of the first seven primes, 2*3*5*7*11*13 = 390390.

			60060 is a term because 60060 = 2^2*3*5*7*11*13 with six distinct prime divisors 2, 3, 5, 7, 11, 13
87780 is a term because 87780 = 2^2*3*5*7*11*19 with six distinct prime divisors 2, 3, 5, 7, 11, 19.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Row 6 of A125666.

Cf. A067885, A001358, A014612, A014613, A014614.

Select[Range[0,5*8! ],Length[FactorInteger[ # ]]==6&] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)

is(n)=omega(n)==6 \\ Charles R Greathouse IV, Jun 19 2016

A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)
list(lim,pr=6)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

{n : A001221(n) = 6} . - R. J. Mathar, Jul 07 2012

A112428 Carmichael numbers equal to the product of 5 primes.

Original entry on oeis.org

825265, 1050985, 9890881, 10877581, 12945745, 13992265, 16778881, 18162001, 27336673, 28787185, 31146661, 36121345, 37167361, 40280065, 41298985, 41341321, 41471521, 47006785, 67371265, 67994641, 69331969, 74165065

Offset: 1

Shyam Sunder Gupta, Dec 11 2005

A subsequence is given by (6n+1)*(12n+1)*(18n+1)*(36n+1)*(72n+1) with n in A206349. - M. F. Hasler, Apr 14 2015

			a(1)=825265=5*7*17*19*73

R. J. Mathar, Table of n, a(n) for n = 1..14938

Cf. A002997, A014614, A206349.

Cf. A087788, A141711, A074379, A112429 - A112432, A006931.

is_A112428(n)=bigomega(n)==5&&is_A002997(n) \\ M. F. Hasler, Apr 14 2015

A112428 = A002997 intersect A014614. - M. F. Hasler, Apr 14 2015

Crossrefs added by M. F. Hasler, Apr 14 2015

A110229 5-almost primes p * q * r * s * t relatively prime to p + q + r + s + t.

Original entry on oeis.org

48, 80, 108, 112, 176, 208, 252, 272, 300, 304, 368, 405, 420, 464, 468, 496, 500, 567, 592, 656, 660, 675, 684, 688, 752, 848, 891, 924, 944, 976, 980, 1020, 1053, 1072, 1116, 1136, 1140, 1168, 1264, 1300, 1323, 1328, 1332, 1372, 1377, 1424, 1428, 1452

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t are not necessarily distinct. The converse to this is A110230: 5-almost primes p * q * r * s * t which are not relatively prime to p+q+r+s+t. A014614 is the 5-almost primes.

			48 is in this sequence because 48 = 2^4 * 3, which has no factors in common with 2 + 2 + 2 + 2 + 3 = 11.

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

Cf. A014614, A002033, A110187, A110188, A110227, A110228, A110230, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\16, forprime(q=2, min(p, lim\8\p), my(pq=p*q); forprime(r=2, min(lim\pq\4, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\2, r), my(pqrs=pqr*s,n); forprime(t=2,min(lim\pqrs,s), n=pqrs*t; if(gcd(n, p+q+r+s+t)==1, listput(v, n))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

Incorrect formula and comment of Sep 2009 related to A002033 deleted - R. J. Mathar, Oct 14 2009

A110230 5-almost primes p * q * r * s * t not relatively prime to p + q + r + s + t.

Original entry on oeis.org

32, 72, 120, 162, 168, 180, 200, 243, 264, 270, 280, 312, 378, 392, 396, 408, 440, 450, 456, 520, 552, 588, 594, 612, 616, 630, 680, 696, 700, 702, 728, 744, 750, 760, 780, 828, 882, 888, 918, 920, 945, 952, 968, 984, 990, 1026, 1032, 1044, 1050, 1064, 1092

Offset: 1

Jonathan Vos Post, Jul 17 2005

p, q, r, s, t are not necessarily distinct. The converse to this is A110229: 5-almost primes p * q * r * s * t which are relatively prime to p+q+r+s+t. A014614 is the 5-almost primes.

			180 is in this sequence because 180 = 2^2 * 3^2 * 5, the sum of the prime factors being 2 + 2 + 3 + 3 + 5 = 15 = 3 * 5 which has two prime factors in common with 180.

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

Cf. A014614, A110187, A110188, A110227, A110228, A110229, A110231, A110232, A110289, A110290, A110296, A110297.

list(lim)=my(v=List()); forprime(p=2, lim\16, forprime(q=2, min(p, lim\8\p), my(pq=p*q); forprime(r=2, min(lim\pq\4, q), my(pqr=pq*r); forprime(s=2, min(lim\pqr\2, r), my(pqrs=pqr*s,n); forprime(t=2,min(lim\pqrs,s), n=pqrs*t; if(gcd(n, p+q+r+s+t)>1, listput(v, n))))))); Set(v) \\ Charles R Greathouse IV, Jan 31 2017

Extended by Ray Chandler, Jul 20 2005

A109018 Least number with exactly n prime factors counted with multiplicity which gives a different number with exactly n prime factors counted with multiplicity when digits are reversed.

Original entry on oeis.org

13, 15, 117, 126, 270, 2576, 8820, 16560, 21168, 46848, 295245, 441600, 846720, 4078080, 80663040, 40590720, 2173236480, 4011724800, 21122906112, 40915058688, 274148425728, 63769149440, 2707602702336, 6167442456576, 21586195906560, 29798871072768, 420127895977984, 631722992467968

Offset: 1

Jonathan Vos Post, Jun 16 2005

An emirp ("prime" spelled backwards) is a prime whose (base 10) reversal is also prime, but which is not a palindromic prime. The first few are 13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, ... (A006567).
An emirpimes ("semiprime" spelled backwards) is a semiprime whose (base 10) reversal is a different semiprime. A list of the first emirpimeses (or "semirpimes") are 15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 122, 123, ... (A097393).
An "emirp tsomla-3" ("3-almost prime" spelled backwards) is the k=3 sequence of the series for which k=1 are emirps and k=2 are emirpimes, a list of these being A109023. The union of these for k=1 through k = 13 is A109019.
The primes correspond to the "1-almost prime" numbers 2, 3, 5, 7, 11, ... (A000040). The 2-almost prime numbers correspond to semiprimes 4, 6, 9, 10, 14, 15, 21, 22, ... (A001358).
The first few 3-almost primes are 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, ... (A014612). The first few 4-almost primes are 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, ... (A014613).
The first few 5-almost primes are 32, 48, 72, 80, ... (A014614).
The Mathematica code for this was written by Ray Chandler, who has coauthorship credit for this sequence.

			a(1) = 13 because 13 is the smallest "emirp" (prime which, digits reversed, becomes a different prime) since reverse(13) = 31 is prime.
a(2) = 15 because 15 is the smallest emirpimes ("semiprime" spelled backwards) as a semiprime whose (base 10) reversal is a different semiprime. The first such number is 15, since 15 reversed is 51 and both 15 and 51 are semiprimes (i.e. 15 = 3 * 5 and 51 = 3 * 17).
a(3) = 117 because 117 is the smallest "emirp tsomla-3" ("3-almost prime" spelled backwards) since 117 reversed is 711 and 117 = 3^2 * 13 and 711 = 3^2 * 79.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A006567, A097393, A109023, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109030, A109031.

kAlmost[n_] := Plus @@ Last /@ FactorInteger@n; fQ[n_] := Block[{id = IntegerDigits@n, k = kAlmost@n}, If[id != Reverse@id && k == kAlmost@FromDigits@Reverse@id, k, -1]]; t = Table[0, {20}]; Do[ a = fQ@n; If[a < 20 && t[[a]] == 0, t[[a]] = n; Print[{a, n}]], {n, 10, 150000000}] (* Robert G. Wilson v, Jan 06 2008 *)
Table[Select[Range[41*10^5],!PalindromeQ[#]&&PrimeOmega[#]==PrimeOmega[ IntegerReverse[ #]] ==n&][[1]],{n,14}] (* The program generates the first 14 terms of the sequence. *) (* Harvey P. Dale, Oct 15 2023 *)

a(14)-a(16) from Robert G. Wilson v, Jan 06 2008
a(17)-a(24) from Donovan Johnson, Nov 17 2008
a(25)-a(28) from Michael S. Branicky, Jun 04 2024

cp's OEIS Frontend

A078841 Main diagonal of the table of k-almost primes (A078840): a(n) = (n+1)-st integer that is an n-almost prime.

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A101638 Number of distinct 4-almost primes A014613 which are factors of n.

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

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

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A058933 Let k be bigomega(n) (i.e., n is a k-almost-prime). a(n) = number of k-almost-primes <= n.

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A074969 Numbers with six distinct prime divisors.

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A112428 Carmichael numbers equal to the product of 5 primes.

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