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Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).
6

%I #6 Jul 28 2016 22:43:52

%S 30,42,66,70,78,102,105,110,114,130,138,154,165,170,174,180,182,186,

%T 190,195,222,230,231,238,246,252,255,258,266,273,282,285,286,290,300,

%U 310,318,322,345,354,357,366,370,374,385,396,399,402,406,410,418,426,429

%N Numbers with prime signatures (1,1,1) or (2,2,1) or (3,2,2).

%C Union of A007304, A179643 and A179695; subsequence of A033992;

%C A001221(a(n)) = 3 and A051903(a(n)) <= A051904(a(n)) + 1 and A001222(a(n)) = 3 or 5 or 7;

%C A050326(a(n)) = 5.

%H Reinhard Zumkeller, <a href="/A225228/b225228.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ 2n log n / (log log n)^2. - _Charles R Greathouse IV_, Jul 28 2016

%e A007304(1) = 2*3*5 = 30, A206778(30,1..8)=[1,2,3,5,6,10,15,30]:

%e A050326(30) = #{30, 15*2, 10*3, 6*5, 5*3*2} = 5;

%e A179643(1) = 2^2*3^2*5 = 180, A206778(180,1..8)=[1,2,3,5,6,10,15,30]:

%e A050326(180) = #{30*6, 30*3*2, 15*6*2, 10*6*3, 6*5*3*2} = 5;

%e A179695(1) = 2^3*3^2*5^2 = 1800, A206778(1800,1..8)=[1,2,3,5,6,10,15,30]:

%e A050326(1800) = #{30*10*6, 30*6*5*2, 30*10*3*2, 15*10*6*2, 10*6*5*3*2} = 5.

%o (Haskell)

%o a225228 n = a225228_list !! (n-1)

%o a225228_list = filter f [1..] where

%o f x = length es == 3 && sum es `elem` [3,5,7] &&

%o maximum es - minimum es <= 1

%o where es = a124010_row x

%o (PARI) is(n)=my(f=vecsort(factor(n)[,2]~)); f==[1,1,1] || f==[1,2,2] || f==[2,2,3] \\ _Charles R Greathouse IV_, Jul 28 2016

%Y Cf. A124010.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, May 03 2013