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A067259 Cubefree numbers which are not squarefree. 24
4, 9, 12, 18, 20, 25, 28, 36, 44, 45, 49, 50, 52, 60, 63, 68, 75, 76, 84, 90, 92, 98, 99, 100, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 169, 171, 172, 175, 180, 188, 196, 198, 204, 207, 212, 220, 225, 228 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
a(n)=m iff A051903(m)=2.
Let us introduce a function D(n)=sigma_0(n)/(2^(alpha(1)+...+alpha(r)), sigma_0(n) number of divisors of n (A000005), prime factorization of n=p(1)^alpha(1) * ... * p(r)^alpha(r), alpha(1)+...+alpha(r) is sequence (A086436). This function splits the set of positive integers into subsets, according to the value of D(n). Squarefree numbers (005117) has D(n)=1, other numbers are "deviated" from the squarefree ideal and have 0 < D(n) < 1. So for D(n)=1/2 we have A048109, D(n)=3/4 we have A067295. - Ctibor O. Zizka, Sep 21 2008
The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... - Amiram Eldar, Jul 09 2020
LINKS
Eric Weisstein's World of Mathematics, Cubefree
Eric Weisstein's World of Mathematics, Squarefree
FORMULA
A212793(a(n)) * (1 - A008966(a(n))) = 1. - Reinhard Zumkeller, May 27 2012
MATHEMATICA
f[n_]:=Union[Last/@FactorInteger[n]][[ -1]]; lst={}; Do[If[f[n]==2, AppendTo[lst, n]], {n, 2, 6!}]; lst (* Vladimir Joseph Stephan Orlovsky, Feb 12 2010 *)
Select[Range[500], Not[SquareFreeQ[#]] && FreeQ[FactorInteger[#], {_, k_ /; k>2}]&] (* Vaclav Kotesovec, Jul 09 2020 *)
PROG
(Haskell)
a067259 n = a067259_list !! (n-1)
a067259_list = filter ((== 2) . a051903) [1..]
-- Reinhard Zumkeller, May 27 2012
(PARI) is(n)=n>3 && vecmax(factor(n)[, 2])==2 \\ Charles R Greathouse IV, Oct 15 2015
CROSSREFS
Sequence in context: A336594 A304365 A038109 * A349931 A060687 A286228
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Feb 20 2002
STATUS
approved

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Last modified February 29 16:58 EST 2024. Contains 370426 sequences. (Running on oeis4.)