A069279 -id:A069279 - cp's OEIS frontend

A069280 19-almost primes (generalization of semiprimes).

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

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

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.

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.

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

A114989 Numbers whose sum of squares of distinct prime factors is prime.

Original entry on oeis.org

6, 10, 12, 14, 18, 20, 24, 26, 28, 34, 36, 40, 48, 50, 52, 54, 56, 68, 72, 74, 80, 94, 96, 98, 100, 104, 105, 108, 112, 134, 136, 144, 146, 148, 160, 162, 188, 192, 194, 196, 200, 206, 208, 216, 224, 231, 250, 268, 272, 273, 274, 288, 292, 296, 315, 320, 324, 326

Offset: 1

Jonathan Vos Post, Feb 22 2006

A005063 is "sum of squares of primes dividing n." Hence this is the sum of squares of prime factors analog of A114522 "numbers n such that sum of distinct prime divisors of n is prime." Note the distinction between A005063 and A067666 is "sum of squares of prime factors of n (counted with multiplicity)."

			a(1) = 6 because 6 = 2 * 3 and 2^2 + 3^2 = 13 is prime.
a(2) = 10 because 10 = 2 * 5 and 2^2 + 5^2 = 29 is prime.
a(3) = 12 because 12 = 2^2 * 3 and 2^2 + 3^2 = 13 is prime (note that we are not counting the prime factors with multiplicity).
a(4) = 14 because 14 = 2 * 7 and 2^2 + 7^2 = 53 is prime.
a(8) = 26 because 26 = 2 * 3 and 2^2 + 13^2 = 173 is prime.
a(10) = 34 because 34 = 2 * 17 and 2^2 + 17^2 = 293 is prime.

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

Cf. A005063, A067666, A014612, A014613, A069273, A069279, A069281, A114522.

with(numtheory): a:=proc(n) local DPF: DPF:=factorset(n): if isprime(sum(DPF[j]^2,j=1..nops(DPF)))=true then n else fi end: seq(a(n),n=1..400); # Emeric Deutsch, Mar 07 2006

Select[Range[400],PrimeQ[Total[Transpose[FactorInteger[#]][[1]]^2]]&] (* Harvey P. Dale, Jan 16 2016 *)

PARI
```
is(n)=isprime(norml2(factor(n)[,1]))
```

{k such that A005063(k) is prime}. {k such that A005063(k) is an element of A000040}. {k = (for distinct i, j, ... prime(i)^e_1 * prime(j)^e_2 * ...) such that (prime(i)^2 * prime(j)^2 * ...) is prime}.

More terms from Emeric Deutsch, Mar 07 2006

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

A114987 Numbers with a 3-almost prime number of prime divisors (counted with multiplicity).

Original entry on oeis.org

256, 384, 576, 640, 864, 896, 960, 1296, 1344, 1408, 1440, 1600, 1664, 1944, 2016, 2112, 2160, 2176, 2240, 2400, 2432, 2496, 2916, 2944, 3024, 3136, 3168, 3240, 3264, 3360, 3520, 3600, 3648, 3712, 3744, 3968, 4000, 4096, 4160, 4374, 4416, 4536, 4704

Offset: 1

Jonathan Vos Post, Feb 22 2006

This is the 3-almost prime analog of A063989 "numbers with a prime number of prime divisors (counted with multiplicity)" and A110893 "numbers with a semiprime number of prime divisors (counted with multiplicity)." Below 4096, this is identical to 8-almost primes (A014613). Between 4096 and 6144, this is identical to 8-almost primes. Below 262144 this is identical to the union of 8-almost primes (A014613) and 12-almost primes (A069273). Between 262144 and 393216, this is identical to the union of 8-almost primes and 12-almost primes.

			a(1) = 256 because 256 = 2^8, which has a 3-almost prime (8) number of prime factors with multiplicity.
a(38) = 4096 because 4096 = 2^12, which has a 3-almost prime (12) number of prime factors with multiplicity.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001222, A014612, A014613, A069273, A069279, A069281.

Select[Range[5000],PrimeOmega[PrimeOmega[#]]==3&] (* Harvey P. Dale, Apr 12 2015 *)

is(n)=bigomega(bigomega(n))==3 \\ Charles R Greathouse IV, Feb 05 2017

a(n) such that A001222(A001222(a(n))) = 3. a(n) such that A001222(a(n)) is an element of A014612. a(n) such that bigomega(a(n)) is an element of A014612. Union[8-almost primes (A014613), 12-almost primes (A069273), 18-almost primes (A069279), 20-almost primes (A069281), 27-almost primes]...

A345740 a(n) is the least prime p such that Omega(p + n) = n where Omega is A001222, or 0 if no such prime exists.

Original entry on oeis.org

2, 2, 5, 131, 43, 15619, 281, 6553, 503, 137771, 3061, 244140613, 8179, 22143361, 401393, 199290359, 491503, 8392333984357, 524269, 3486784381, 2097131, 226640986043, 28311529, 303745269775390601, 113246183, 9885033776809, 469762021, 176518460300597, 805306339, 77737724676061053405079339

Offset: 1

Zak Seidov, Jun 25 2021

nonn

			For n=1, a(1) = 2 as 2+1 = 3 (Omega(2 + 1) = Omega(3) = 1, see A000040(1)).
For n=2, 2+2 = 4 = 2*2 (semiprime, Omega(4) = 2, see A001358(1)).
For n=3, 5+3 = 8 = 2*2*2 (triprime, Omega(8) = 3, see A014612(1)).
For n=4, 131+4 = 135 = 3*3*3*5 (Omega(135) = 4,  see A014613(16)).

David A. Corneth, Table of n, a(n) for n = 1..1049

Cf. A000040, A001222, A001358, A014612, A014613, A053669, A069279, A097977.

Table[k=1;While[PrimeOmega[Prime@k+n]!=n,k++];Prime@k,{n,11}] (* Giorgos Kalogeropoulos, Jun 25 2021 *)

a(n) = my(p=2); while (bigomega(p+n) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jun 26 2021

from sympy import factorint, nextprime, primerange
def Omega(n): return sum(e for f, e in factorint(n).items())
def a(n):
    lb = 2**n
    p = nextprime(max(lb-n, 1) - 1)
    while Omega(p+n) != n: p = nextprime(p)
    return p
print([a(n) for n in range(1, 12)]) # Michael S. Branicky, Aug 14 2021

a(n) + n >= A053669(n)^n for n > 2 if a(n) exists. - David A. Corneth, Aug 14 2021

A123118 Partial products of A101695.

Original entry on oeis.org

2, 12, 216, 8640, 933120, 209018880, 100329062400, 130026464870400, 349511137571635200, 1968446726803449446400, 22676506292775737622528000, 522466704985552994823045120000, 27820307107070725868337506549760000

Offset: 1

Jonathan Vos Post, Sep 28 2006

nonn

The number of prime factors (with multiplicity) of a(n) is T(n) = A000217(n) = n*(n+1)/2.

			a(1) = 2 = prime(1).
a(2) = 12 = 2 * 6 = prime(1) * semiprime(2) = 2^2 * 3.
a(3) = 216 = 2 * 6 * 18 = prime(1) * semiprime(2) * 3-almostprime(3) = 2^3 * 3^3.
a(4) = 8640 = 2 * 6 * 18 * 40 = prime(1) * semiprime(2) * 3-almostprime(3) * 4-almostprime(4) = 2^6 * 3^3 * 5.
a(15) = 893179304874387947794472921245209518407680000 = 2 * 6 * 18 * 40 * 108 * 224 * 480 * 1296 * 2688 * 5632 * 11520 * 23040 * 53248 * 124416 * 258048 = 2^88 * 3^23 * 5^4 * 7^3 * 11 * 13.

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.

a(n) = Prod[i=1..n] i-th i-almost prime = Prod[i=1..n] A101695(i).

A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

Original entry on oeis.org

4, 50, 189, 1863, 10449, 447849, 4449249, 5745249, 3606422049, 16554218751, 105265530369, 1957645712385

Offset: 2

Zak Seidov, Nov 13 2018

From Jon E. Schoenfield, Nov 15 2018: (Start)
If a(11) is odd, it is 16554218751.
If a(12) is odd, it is 105265530369.
If a(13) is odd, it is 1957645712385. (End)
a(11), a(12), and a(13) are indeed odd. - Giovanni Resta, Jan 04 2019
10^13 < a(14) <= 240455334218751, a(15) <= 2992278212890624. - Giovanni Resta, Jan 06 2019

			For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.

Cf. A078840.

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), A069281 (r = 20).

a[n_] := Module[{o={0,0,0}, k=1}, While[o!={n-1,n,n-1}, o=Rest[AppendTo[o,PrimeOmega[k]]]; k++]; k-2]; Array[a,7,2] (* Amiram Eldar, Nov 14 2018 *)

{for(n=2,10,for(k=2^n,10^12,if(n==bigomega(k) &&
n-1==bigomega(k-1) && n-1==bigomega(k+1),print1(k", ");break())))}

a(10) from Jon E. Schoenfield, Nov 14 2018

a(11)-a(13) from Giovanni Resta, Jan 04 2019

cp's OEIS Frontend

A069280 19-almost primes (generalization of semiprimes).

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

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

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A114989 Numbers whose sum of squares of distinct prime factors is prime.

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

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A114987 Numbers with a 3-almost prime number of prime divisors (counted with multiplicity).

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