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Products of 7 distinct primes (squarefree 7-almost primes).
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%I #27 Aug 31 2024 12:29:16

%S 510510,570570,690690,746130,870870,881790,903210,930930,1009470,

%T 1067430,1111110,1138830,1193010,1217370,1231230,1272810,1291290,

%U 1345890,1360590,1385670,1411410,1438710,1452990,1504230,1540770

%N Products of 7 distinct primes (squarefree 7-almost primes).

%C Intersection of A005117 and A046308.

%C Intersection of A005117 and A176655. - _R. J. Mathar_, Dec 05 2016

%H Rick L. Shepherd, <a href="/A123321/b123321.txt">Table of n, a(n) for n = 1..10000</a>

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

%t 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 Select[Range[1600000],PrimeNu[#]==7&&SquareFreeQ[#]&] (* _Harvey P. Dale_, Sep 19 2013 *)

%o (PARI) is(n)=omega(n)==7 && bigomega(n)==7 \\ _Hugo Pfoertner_, Dec 18 2018

%o (Python)

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A123321(n):

%o 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 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 def bisection(f,kmin=0,kmax=1):

%o while f(kmax) > kmax: kmax <<= 1

%o while kmax-kmin > 1:

%o kmid = kmax+kmin>>1

%o if f(kmid) <= kmid:

%o kmax = kmid

%o else:

%o kmin = kmid

%o return kmax

%o return bisection(f) # _Chai Wah Wu_, Aug 31 2024

%Y 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 nonn

%O 1,1

%A _Rick L. Shepherd_, Sep 25 2006

%E More terms from _Vladimir Joseph Stephan Orlovsky_, Aug 26 2008