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

%I A123321 #27 Aug 31 2024 12:29:16
%S A123321 510510,570570,690690,746130,870870,881790,903210,930930,1009470,
%T A123321 1067430,1111110,1138830,1193010,1217370,1231230,1272810,1291290,
%U A123321 1345890,1360590,1385670,1411410,1438710,1452990,1504230,1540770
%N A123321 Products of 7 distinct primes (squarefree 7-almost primes).
%C A123321 Intersection of A005117 and A046308.
%C A123321 Intersection of A005117 and A176655. - _R. J. Mathar_, Dec 05 2016
%H A123321 Rick L. Shepherd, <a href="/A123321/b123321.txt">Table of n, a(n) for n = 1..10000</a>
%e A123321 a(1) = 510510 = 2*3*5*7*11*13*17 = A002110(7).
%t A123321 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 *)
%t A123321 Select[Range[1600000],PrimeNu[#]==7&&SquareFreeQ[#]&] (* _Harvey P. Dale_, Sep 19 2013 *)
%o A123321 (PARI) is(n)=omega(n)==7 && bigomega(n)==7 \\ _Hugo Pfoertner_, Dec 18 2018
%o A123321 (Python)
%o A123321 from math import isqrt, prod
%o A123321 from sympy import primerange, integer_nthroot, primepi
%o A123321 def A123321(n):
%o A123321     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)))
%o A123321     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)))
%o A123321     def bisection(f,kmin=0,kmax=1):
%o A123321         while f(kmax) > kmax: kmax <<= 1
%o A123321         while kmax-kmin > 1:
%o A123321             kmid = kmax+kmin>>1
%o A123321             if f(kmid) <= kmid:
%o A123321                 kmax = kmid
%o A123321             else:
%o A123321                 kmin = kmid
%o A123321         return kmax
%o A123321     return bisection(f) # _Chai Wah Wu_, Aug 31 2024
%Y A123321 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).
%K A123321 nonn
%O A123321 1,1
%A A123321 _Rick L. Shepherd_, Sep 25 2006
%E A123321 More terms from _Vladimir Joseph Stephan Orlovsky_, Aug 26 2008