cp's OEIS Frontend

A381823 Odd cubefree numbers that are not squarefree.

%I A381823 #8 Mar 09 2025 12:27:44
%S A381823 9,25,45,49,63,75,99,117,121,147,153,169,171,175,207,225,245,261,275,
%T A381823 279,289,315,325,333,361,363,369,387,423,425,441,475,477,495,507,525,
%U A381823 529,531,539,549,575,585,603,605,637,639,657,693,711,725,735,747,765,775
%N A381823 Odd cubefree numbers that are not squarefree.
%C A381823 Numbers whose prime factorization has only odd primes, exponents that are smaller than 3 and at least one exponent that equals 2.
%C A381823 Odd numbers k such that A051903(k) = A375039((k+1)/2) = 2.
%C A381823 The asymptotic density of this sequence is 4/(7*zeta(3)) - 2/(3*zeta(2)) = 0.070090906905338896329... .
%C A381823 In general, the asymptotic density of odd k-free numbers (numbers that are not divisible by a k-th power other than 1) that are not (k-1)-free, for k >= 2, is 2^(k-1)/((2^k-1) * zeta(k)) - 2^(k-2)/((2^(k-1)-1) * zeta(k-1)).
%H A381823 Amiram Eldar, <a href="/A381823/b381823.txt">Table of n, a(n) for n = 1..10000</a>
%t A381823 Select[Range[1, 1000, 2], Max[FactorInteger[#][[;;, 2]]] == 2 &]
%o A381823 (PARI) isok(k) = k % 2 && if(k == 1, 0, vecmax(factor(k)[, 2]) == 2);
%Y A381823 Intersection of A005408 and A067259.
%Y A381823 Complement of A056911 within A381822.
%Y A381823 Cf. A002117, A013661, A051903, A375039.
%Y A381823 Subsequence of A048103.
%K A381823 nonn,easy
%O A381823 1,1
%A A381823 _Amiram Eldar_, Mar 08 2025