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A072588
Numbers having at least one prime factor with an odd and one with an even exponent.
2
12, 18, 20, 28, 44, 45, 48, 50, 52, 60, 63, 68, 72, 75, 76, 80, 84, 90, 92, 98, 99, 108, 112, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 171, 172, 175, 176, 180, 188, 192, 198, 200, 204, 207, 208, 212, 220, 228, 234, 236, 240, 242, 244
OFFSET
1,1
COMMENTS
= Complement(Union(A002035, A000290)) = Intersection(A000037, A072587);
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.
The asymptotic density of this sequence is 1 - A065463 = 0.2955577990... - Amiram Eldar, Sep 18 2022
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
LINKS
MATHEMATICA
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 *)
PROG
(Haskell)
a072588 n = a072588_list !! (n-1)
a072588_list = filter f [1..] where
f x = any odd es && any even es where es = a124010_row x
-- Reinhard Zumkeller, Nov 15 2012
(PARI) is(n)=#Set(factor(n)[, 2]%2)==2 \\ Charles R Greathouse IV, Oct 16 2015
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jun 23 2002
STATUS
approved