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

Author

Reinhard Zumkeller, Jun 23 2002

Keywords

• nonn

Comments

= Complement(Union(A002035, A000290)) = Intersection(A000037, A072587);

a(k)=A070011(k) for 0A070011(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

Crossrefs

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

Cf. A007913, A007947, A336643.

Programs

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

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 *)

PARI

is(n)=#Set(factor(n)[,2]%2)==2 \\ Charles R Greathouse IV, Oct 16 2015