A046308 -id:A046308 - cp's OEIS frontend

A069277 16-almost primes (generalization of semiprimes).

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.

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.

A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

Original entry on oeis.org

288, 800, 972, 1568, 3872, 5408, 6075, 9248, 11552, 11907, 12500, 16928, 26912, 28125, 29403, 30752, 41067, 43808, 53792, 59168, 67228, 70227, 70688, 87723, 89888, 111392, 119072, 128547, 143648, 151263, 153125, 161312, 170528, 199712

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

nonn

288=2^5*3^2, 800=2^5*5^2,..

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A030636, A046308, A007774.

Cf. A085548, A085965, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={2,5}; Select[Range[200000], f]

list(lim)=my(v=List(),t);forprime(p=2,(lim\4)^(1/5),t=p^5;forprime(q=2,sqrt(lim\t),if(p==q,next);listput(v,t*q^2)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A189988(n):
    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 f(x): return n+x-sum(primepi(isqrt(x//p**4)) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,6)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Sum_{n>=1} 1/a(n) = P(2)*P(5) - P(7) = A085548 * A085965 - A085967 = 0.007886..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

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

A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

Original entry on oeis.org

432, 648, 2000, 5000, 5488, 10125, 16875, 19208, 21296, 27783, 35152, 64827, 78608, 107811, 109744, 117128, 177957, 194672, 214375, 228488, 300125, 390224, 395307, 397953, 476656, 555579, 668168, 771147, 810448, 831875

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, List of Prime Signatures
Index to sequences related to prime signature

Cf. A046308, A030638, A007774.

Cf. A085541, A085964, A085967.

f[n_]:=Sort[Last/@FactorInteger[n]]=={3,4}; Select[Range[10^6], f]
With[{nn=40},Select[Flatten[{#[[1]]^4 #[[2]]^3,#[[1]]^3 #[[2]]^4}&/@ Subsets[ Prime[Range[nn]],{2}]]//Union,#<=16nn^3&]] (* Harvey P. Dale, Nov 15 2020 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\8)^(1/4),t=p^4;forprime(q=2,(lim\t)^(1/3),if(p==q,next);listput(v,t*q^3)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, integer_nthroot, primerange
def A179666(n):
    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 f(x): return n+x-sum(primepi(integer_nthroot(x//p**4,3)[0]) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,7)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Sum_{n>=1} 1/a(n) = P(3)*P(4) - P(7) = A085541 * A085964 - A085967 = 0.005171..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A123321 Products of 7 distinct primes (squarefree 7-almost primes).

Original entry on oeis.org

510510, 570570, 690690, 746130, 870870, 881790, 903210, 930930, 1009470, 1067430, 1111110, 1138830, 1193010, 1217370, 1231230, 1272810, 1291290, 1345890, 1360590, 1385670, 1411410, 1438710, 1452990, 1504230, 1540770

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046308.

Intersection of A005117 and A176655. - R. J. Mathar, Dec 05 2016

			a(1) = 510510 = 2*3*5*7*11*13*17 = A002110(7).

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A005117, A046308, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123322 (k=8), A115343 (k=9).

f7Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f7Q[n], AppendTo[lst, n]], {n, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Select[Range[1600000],PrimeNu[#]==7&&SquareFreeQ[#]&] (* Harvey P. Dale, Sep 19 2013 *)

is(n)=omega(n)==7 && bigomega(n)==7 \\ Hugo Pfoertner, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123321(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,0,1,1,7)))
    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

More terms from Vladimir Joseph Stephan Orlovsky, Aug 26 2008

A110289 7-almost primes pqrstuv relatively prime to p+q+r+s+t+u+v.

Original entry on oeis.org

320, 432, 448, 704, 720, 832, 972, 1088, 1216, 1472, 1584, 1680, 1856, 1984, 2000, 2268, 2352, 2368, 2448, 2624, 2700, 2752, 3008, 3120, 3312, 3392, 3645, 3696, 3776, 3904, 3920, 4176, 4212, 4288, 4368, 4400, 4544, 4672, 5056, 5103, 5200, 5312, 5488

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110290, is 7-almost primes p*q*r*s*t*u*v which are not relatively prime to p+q+r+s+t+u+v.

Contains p*q^6 if p and q are distinct primes, p >= 5. - Robert Israel, Jan 13 2017

			832 = 2^6 * 13 is in this sequence because its sum of prime factors is 2 + 2 + 2 + 2 + 2 + 2 + 13 = 25 = 5^2, which has no factor in common with 832.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110290, A110296, A110297.

Cf. A001414 (sopfr(n)).

N:= 10^4: # to get all terms <= N
P:= select(isprime, [$1..N/2^6]):
nP:= nops(P):
Res:= {}:
for p in P do
  for q in P while q <= p and p*q*2^5 <= N do
    for r in P while r <= q and p*q*r*2^4 <= N do
      for s in P while s <= r and p*q*r*s*2^3 <= N do
        for t in P while t <= s and p*q*r*s*t*2^2 <= N do
          for u in P while u <= t and p*q*r*s*t*u*2 <= N do
            for v in P while v <= u and p*q*r*s*t*u*v <= N do
              if igcd(p+q+r+s+t+u+v,p*q*r*s*t*u*v) = 1 then
                  Res:= Res union {p*q*r*s*t*u*v} fi
od od od od od od od:
sort(convert(Res,list)); # Robert Israel, Jan 13 2017

Select[Range[6000],PrimeOmega[#]==7&&CoprimeQ[Total[ Times@@@ FactorInteger[ #]],#]&] (* Harvey P. Dale, Nov 19 2019 *)

sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))
isok(n)=bigomega(n)==7&&gcd(n, sopfr(n))==1 \\ Rick L. Shepherd, Jul 20 2005

Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005

A110290 7-almost primes pqrstuv not relatively prime to p+q+r+s+t+u+v.

Original entry on oeis.org

128, 192, 288, 480, 648, 672, 800, 1008, 1056, 1080, 1120, 1200, 1248, 1458, 1512, 1568, 1620, 1632, 1760, 1800, 1824, 1872, 2080, 2187, 2208, 2376, 2430, 2464, 2520, 2640, 2720, 2736, 2784, 2800, 2808, 2912, 2976, 3000, 3040, 3402, 3528, 3552, 3564

Offset: 1

Jonathan Vos Post, Jul 18 2005

The primes p, q, r, s, t, u, v are not necessarily distinct. The 7-almost primes are A046308. The converse, A110289, is 7-almost primes p*q*r*s*t*u*v which are relatively prime to p+q+r+s+t+u+v.

			800 = 2^5 * 5^2 is in this sequence because the sum of prime factors 2 + 2 + 2 + 2 + 2 + 5 + 5 = 20 is not relatively prime to 800 (in fact, it is a divisor of 800).

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

Cf. A046308, A110187, A110188, A110227, A110228, A110229, A110230, A110231, A110232, A110289, A110296, A110297.

Cf. A001414 (sopfr(n)).

Select[Range[4000],PrimeOmega[#]==7&&!CoprimeQ[Total[Flatten[Table[ #[[1]], #[[2]]]&/@ FactorInteger[#]]],#]&] (* Harvey P. Dale, Apr 30 2018 *)

sopfr(n)=local(f);if(n<1,0,f=factor(n);sum(k=1,matsize(f)[1],f[k,1]*f[k,2]))
isok(n)=bigomega(n)==7&&gcd(n, sopfr(n))>1 \\ Rick L. Shepherd, Jul 20 2005

Extended by Ray Chandler and Rick L. Shepherd, Jul 20 2005

cp's OEIS Frontend

A069277 16-almost primes (generalization of semiprimes).

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A069274 13-almost primes (generalization of semiprimes).

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A069278 17-almost primes (generalization of semiprimes).

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A069280 19-almost primes (generalization of semiprimes).

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A179646 Product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2).

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

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A179666 Products of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3).

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A110289 7-almost primes pqrstuv relatively prime to p+q+r+s+t+u+v.

A110290 7-almost primes pqrstuv not relatively prime to p+q+r+s+t+u+v.