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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

Crossrefs

Cf. A000037, A000290, A001221, A001222, A002035, A065463, A070011, A072587, A124010.

Cf. A007913, A007947, A336643.

Sequence in context: A359892 A070011 A084679 * A267117 A187039 A360554

Adjacent sequences: A072585 A072586 A072587 * A072589 A072590 A072591

Keyword

nonn

Author

Reinhard Zumkeller, Jun 23 2002

Status

approved