|
%I #30 Jul 04 2019 10:27:43
%S 6,8,10,12,14,15,16,18,20,21,22,24,26,27,28,30,32,33,34,35,36,38,39,
%T 40,42,44,45,46,48,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,
%U 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 Numbers having at least two distinct or a total of at least three prime factors.
%C Complement of A000430; A080256(a(n)) > 3.
%C A084114(a(n)) > 0, see also A084110.
%C Also numbers greater than the square of their smallest prime-factor: a(n)>A020639(a(n))^2=A088377(a(n));
%C a(n)>A000430(k) for n<=13, a(n) < A000430(k) for n>13.
%C Numbers with at least 4 divisors. - _Franklin T. Adams-Watters_, Jul 28 2006
%C Union of A024619 and A033942; A211110(a(n)) > 2. - _Reinhard Zumkeller_, Apr 02 2012
%C 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
%H Reinhard Zumkeller, <a href="/A080257/b080257.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = n + O(n/log n). - _Charles R Greathouse IV_, Sep 14 2015
%e 8=2*2*2 and 10=2*5 are terms; 4=2*2 is not a term.
%e From _Gus Wiseman_, Jul 03 2019: (Start)
%e The sequence of terms together with their prime indices begins:
%e 6: {1,2}
%e 8: {1,1,1}
%e 10: {1,3}
%e 12: {1,1,2}
%e 14: {1,4}
%e 15: {2,3}
%e 16: {1,1,1,1}
%e 18: {1,2,2}
%e 20: {1,1,3}
%e 21: {2,4}
%e 22: {1,5}
%e 24: {1,1,1,2}
%e 26: {1,6}
%e 27: {2,2,2}
%e 28: {1,1,4}
%e 30: {1,2,3}
%e 32: {1,1,1,1,1}
%e (End)
%t Select[Range[100],PrimeNu[#]>1||PrimeOmega[#]>2&] (* _Harvey P. Dale_, Jul 23 2013 *)
%o (Haskell)
%o a080257 n = a080257_list !! (n-1)
%o a080257_list = m a024619_list a033942_list where
%o m xs'@(x:xs) ys'@(y:ys) | x < y = x : m xs ys'
%o | x == y = x : m xs ys
%o | x > y = y : m xs' ys
%o -- _Reinhard Zumkeller_, Apr 02 2012
%o (PARI) is(n)=omega(n)>1 || isprimepower(n)>2
%Y Cf. A001248, A001221, A001222, A006881, A030078, A088381, A088383, A000005.
%Y Cf. A060687, A118914, A323014, A323023, A325249.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, Feb 10 2003
%E Definition clarified by _Harvey P. Dale_, Jul 23 2013
|
