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A069278 17-almost primes (generalization of semiprimes).

%I A069278 #36 Feb 23 2025 14:09:41
%S A069278 131072,196608,294912,327680,442368,458752,491520,663552,688128,
%T A069278 720896,737280,819200,851968,995328,1032192,1081344,1105920,1114112,
%U A069278 1146880,1228800,1245184,1277952,1492992,1507328,1548288,1605632,1622016
%N A069278 17-almost primes (generalization of semiprimes).
%C A069278 Product of 17 not necessarily distinct primes.
%C A069278 Divisible by exactly 17 prime powers (not including 1).
%C A069278 For n = 1..2628 a(n)=2*A069277(n). - _Zak Seidov_, Jun 25 2017
%H A069278 D. W. Wilson, <a href="/A069278/b069278.txt">Table of n, a(n) for n = 1..10000</a>
%H A069278 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F A069278 Product p_i^e_i with Sum e_i = 17.
%t A069278 Select[Range[2*10^6],PrimeOmega[#]==17&] (* _Harvey P. Dale_, Sep 28 2016 *)
%o A069278 (PARI)
%o A069278 k=17; start=2^k; finish=2000000; v=[]
%o A069278 for(n=start,finish, if(bigomega(n)==k,v=concat(v,n))); v
%o A069278 (Python)
%o A069278 from math import isqrt, prod
%o A069278 from sympy import primerange, integer_nthroot, primepi
%o A069278 def A069278(n):
%o A069278     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)))
%o A069278     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)))
%o A069278     def bisection(f,kmin=0,kmax=1):
%o A069278         while f(kmax) > kmax: kmax <<= 1
%o A069278         while kmax-kmin > 1:
%o A069278             kmid = kmax+kmin>>1
%o A069278             if f(kmid) <= kmid:
%o A069278                 kmax = kmid
%o A069278             else:
%o A069278                 kmin = kmid
%o A069278         return kmax
%o A069278     return bisection(f) # _Chai Wah Wu_, Aug 31 2024
%Y A069278 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
%K A069278 nonn
%O A069278 1,1
%A A069278 _Rick L. Shepherd_, Mar 13 2002