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A072587 Numbers having at least one prime factor with an even exponent.

%I A072587 #55 May 31 2024 21:03:23
%S A072587 4,9,12,16,18,20,25,28,36,44,45,48,49,50,52,60,63,64,68,72,75,76,80,
%T A072587 81,84,90,92,98,99,100,108,112,116,117,121,124,126,132,140,144,147,
%U A072587 148,150,153,156,162,164,169,171,172,175,176,180,188,192,196,198,200,204
%N A072587 Numbers having at least one prime factor with an even exponent.
%C A072587 Complement of the union of {1} and A002035. [Correction, Nov 15 2012]
%C A072587 A162645 is a subsequence and this sequence is a subsequence of A162643. - _Reinhard Zumkeller_, Jul 08 2009
%C A072587 The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - _Amiram Eldar_, Jul 21 2020
%C A072587 A number k is a term iff its core (A007913) properly divides its kernel (A007947), that is iff A336643(k) > 1. - _David James Sycamore_, Sep 18 2023
%H A072587 Reinhard Zumkeller, <a href="/A072587/b072587.txt">Table of n, a(n) for n = 1..10000</a>
%t A072587 Select[Range[210], MemberQ[EvenQ[Transpose[FactorInteger[#]][[2]]], True] &] (* _Harvey P. Dale_, Apr 03 2012 *)
%o A072587 (Haskell)
%o A072587 a072587 n = a072587_list !! (n-1)
%o A072587 a072587_list = tail $ filter (any even . a124010_row) [1..]
%o A072587 -- _Reinhard Zumkeller_, Nov 15 2012
%o A072587 (PARI) is(n)=n>3 && Set(factor(n)[,2]%2)[1]==0 \\ _Charles R Greathouse IV_, Oct 16 2015
%o A072587 (Python)
%o A072587 from itertools import count, islice
%o A072587 from sympy import factorint
%o A072587 def A072587_gen(startvalue=1): # generator of terms
%o A072587     return filter(lambda n:not all(map(lambda m:m&1,factorint(n).values())),count(max(startvalue,1)))
%o A072587 A072587_list = list(islice(A072587_gen(),30)) # _Chai Wah Wu_, Jan 04 2023
%Y A072587 Cf. A000037, A000203, A002035, A065463, A072588, A124010, A162643, A162645, A188999.
%Y A072587 Cf. A007913, A007947, A336643.
%K A072587 nonn
%O A072587 1,1
%A A072587 _Reinhard Zumkeller_, Jun 23 2002
%E A072587 Thanks to _Zak Seidov_, who noticed that 1 had to be removed. - _Reinhard Zumkeller_, Nov 15 2012