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A105441 Numbers with at least two odd prime factors (not necessarily distinct).

%I A105441 #31 Feb 02 2025 19:59:25
%S A105441 9,15,18,21,25,27,30,33,35,36,39,42,45,49,50,51,54,55,57,60,63,65,66,
%T A105441 69,70,72,75,77,78,81,84,85,87,90,91,93,95,98,99,100,102,105,108,110,
%U A105441 111,114,115,117,119,120,121,123,125,126,129,130,132,133,135,138,140,141
%N A105441 Numbers with at least two odd prime factors (not necessarily distinct).
%C A105441 Also polite numbers (A138591) that can be expressed as the sum of two or more consecutive integers in more than one ways. For example 9=4+5 and 9=2+3+4. Also 15=7+8, 15=4+5+6 and 15=1+2+3+4+5. - _Jayanta Basu_, Apr 30 2013
%H A105441 Charles R Greathouse IV, <a href="/A105441/b105441.txt">Table of n, a(n) for n = 1..10000</a>
%F A105441 A087436(a(n)) > 1.
%F A105441 A001227(a(n)) > 2. [_Reinhard Zumkeller_, May 01 2012]
%t A105441 opf3Q[n_]:=Count[Flatten[Table[First[#],{Last[#]}]&/@FactorInteger[n]], _?OddQ]>1 (* _Harvey P. Dale_, Jun 13 2011 *)
%o A105441 (PARI) upTo(lim)=my(v=List(),p=7,m);forprime(q=8,lim,forstep(n=p+2,q-2,2,m=n;while(m<=lim,listput(v,m);m<<=1));p=q);forstep(n=p+2,lim,2,listput(v,n));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Aug 08 2011
%o A105441 (PARI) is(n)=n>>=valuation(n,2); !isprime(n) && n>1 \\ _Charles R Greathouse IV_, Apr 30 2013
%o A105441 (Haskell)
%o A105441 a105441 n = a105441_list !! (n-1)
%o A105441 a105441_list = filter ((> 2) . a001227) [1..]
%o A105441 -- _Reinhard Zumkeller_, May 01 2012
%o A105441 (Python)
%o A105441 from sympy import primepi
%o A105441 def A105441(n):
%o A105441     def f(x): return int(n+1+sum(primepi(x>>i) for i in range(x.bit_length())))
%o A105441     m, k = n, f(n)
%o A105441     while m != k: m, k = k, f(k)
%o A105441     return m # _Chai Wah Wu_, Feb 02 2025
%Y A105441 Complement of A093641; A093642 is a subsequence.
%Y A105441 Cf. A001227, A087436, A138591.
%K A105441 nonn,easy
%O A105441 1,1
%A A105441 _Reinhard Zumkeller_, Apr 09 2005