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A080257 Numbers having at least two distinct or a total of at least three prime factors.

%I A080257 #35 Jan 23 2025 15:02:59
%S A080257 6,8,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,
%T A080257 40,42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,
%U A080257 72,74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94,95,96,98,99,100
%N A080257 Numbers having at least two distinct or a total of at least three prime factors.
%C A080257 Complement of A000430; A080256(a(n)) > 3.
%C A080257 A084114(a(n)) > 0, see also A084110.
%C A080257 Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
%C A080257 a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
%C A080257 Numbers with at least 4 divisors. - _Franklin T. Adams-Watters_, Jul 28 2006
%C A080257 Union of A024619 and A033942; A211110(a(n)) > 2. - _Reinhard Zumkeller_, Apr 02 2012
%C A080257 Also numbers > 1 that are neither prime nor a square of a prime. Also numbers whose omega-sequence (A323023) has sum > 3. Numbers with omega-sequence summing to m are: A000040 (m = 1), A001248 (m = 3), A030078 (m = 4), A068993 (m = 5), A050997 (m = 6), A325264 (m = 7). - _Gus Wiseman_, Jul 03 2019
%C A080257 Numbers n such that sigma_2(n)*tau(n) = A001157(n)*A000005(n) >= 4*n^2. Note that sigma_2(n)*tau(n) >= sigma(n)^2 = A072861 for all n. - _Joshua Zelinsky_, Jan 23 2025
%H A080257 Reinhard Zumkeller, <a href="/A080257/b080257.txt">Table of n, a(n) for n = 1..10000</a>
%F A080257 a(n) = n + O(n/log n). - _Charles R Greathouse IV_, Sep 14 2015
%e A080257 8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
%e A080257 From _Gus Wiseman_, Jul 03 2019: (Start)
%e A080257 The sequence of terms together with their prime indices begins:
%e A080257    6: {1,2}
%e A080257    8: {1,1,1}
%e A080257   10: {1,3}
%e A080257   12: {1,1,2}
%e A080257   14: {1,4}
%e A080257   15: {2,3}
%e A080257   16: {1,1,1,1}
%e A080257   18: {1,2,2}
%e A080257   20: {1,1,3}
%e A080257   21: {2,4}
%e A080257   22: {1,5}
%e A080257   24: {1,1,1,2}
%e A080257   26: {1,6}
%e A080257   27: {2,2,2}
%e A080257   28: {1,1,4}
%e A080257   30: {1,2,3}
%e A080257   32: {1,1,1,1,1}
%e A080257 (End)
%t A080257 Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* _Harvey P. Dale_, Jul 23 2013 *)
%o A080257 (Haskell)
%o A080257 a080257 n = a080257_list !! (n-1)
%o A080257 a080257_list = m a024619_list a033942_list where
%o A080257    m xs'@(x:xs) ys'@(y:ys) | x < y  = x : m xs ys'
%o A080257                            | x == y = x : m xs ys
%o A080257                            | x > y  = y : m xs' ys
%o A080257 -- _Reinhard Zumkeller_, Apr 02 2012
%o A080257 (PARI) is(n)=omega(n)>1 || isprimepower(n)>2
%o A080257 (PARI) is(n)=my(k=isprimepower(n)); if(k, k>2, !isprime(n)) \\ _Charles R Greathouse IV_, Jan 23 2025
%Y A080257 Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.
%Y A080257 Cf. A060687, A118914, A323014, A323023, A325249.
%K A080257 nonn,easy
%O A080257 1,1
%A A080257 _Reinhard Zumkeller_, Feb 10 2003
%E A080257 Definition clarified by _Harvey P. Dale_, Jul 23 2013