cp's OEIS Frontend

A072588 Numbers having at least one prime factor with an odd and one with an even exponent.

%I A072588 #20 Sep 25 2023 08:56:48
%S A072588 12,18,20,28,44,45,48,50,52,60,63,68,72,75,76,80,84,90,92,98,99,108,
%T A072588 112,116,117,124,126,132,140,147,148,150,153,156,162,164,171,172,175,
%U A072588 176,180,188,192,198,200,204,207,208,212,220,228,234,236,240,242,244
%N A072588 Numbers having at least one prime factor with an odd and one with an even exponent.
%C A072588 = Complement(Union(A002035, A000290)) = Intersection(A000037, A072587);
%C A072588 a(k)=A070011(k) for 0<k<=25, A070011(26)=120 is not a term, as 120=5*3*2^3 having only odd exponents (A002035(85)=120), and a(54)=240 is not a term of A070011, as from 240=5*3*2^4 follows that A001222(240)/A001221(240)=6/3=2 is an integer.
%C A072588 The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - _Amiram Eldar_, Sep 18 2022
%C A072588 Numbers k such that A007913(k) properly divides A007947(k). (Same as A072587 without square terms). A number k is in this sequence iff 1 < A007913(k) < A007947(k) < k, and A007913(k)|A007947(k), equivalently iff k is not in A000037 and A336643(k) is squarefree. - _David James Sycamore_, Sep 20 2023
%H A072588 Reinhard Zumkeller, <a href="/A072588/b072588.txt">Table of n, a(n) for n = 1..10000</a>
%t A072588 oeeQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[2]]},Count[fi,_?OddQ]>0  && Count[fi,_?EvenQ]>0]; Select[Range[250],oeeQ] (* _Harvey P. Dale_, Jun 21 2015 *)
%o A072588 (Haskell)
%o A072588 a072588 n = a072588_list !! (n-1)
%o A072588 a072588_list = filter f [1..] where
%o A072588    f x = any odd es && any even es  where es = a124010_row x
%o A072588 -- _Reinhard Zumkeller_, Nov 15 2012
%o A072588 (PARI) is(n)=#Set(factor(n)[,2]%2)==2 \\ _Charles R Greathouse IV_, Oct 16 2015
%Y A072588 Cf. A000037, A000290, A001221, A001222, A002035, A065463, A070011, A072587, A124010.
%Y A072588 Cf. A007913, A007947, A336643.
%K A072588 nonn
%O A072588 1,1
%A A072588 _Reinhard Zumkeller_, Jun 23 2002