cp's OEIS Frontend

A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).

%I A384520 #13 Jun 01 2025 09:58:32
%S A384520 8,24,27,32,40,54,56,88,96,104,120,125,128,135,136,152,160,168,184,
%T A384520 189,216,224,232,243,248,250,264,270,280,296,297,312,328,343,344,351,
%U A384520 352,375,376,378,384,408,416,424,440,456,459,472,480,486,488,512,513,520,536
%N A384520 Numbers whose powerful part (A057521) is greater than 1 and is equal to a squarefree number raised to an odd power (A384518).
%C A384520 Subsequence of A301517 and A374459 and first differs from them at n = 85: A374459(85) = A374459(85) = 864 = 2^5 * 3^3 is not a term of this sequence.
%C A384520 First differs from its subsequence A381312 at n = 21: a(21) = 216 = 2^3 * 3^3 is not a term of A381312.
%C A384520 Numbers whose prime factorization has one distinct exponent that is larger than 1 and it is odd.
%C A384520 Numbers that are a product of a squarefree number (A005117) and a coprime nonsquarefree number that is a squarefree number raised to an odd power (A384518).
%C A384520 The asymptotic density of this sequence is Sum_{k>=1} (d(2*k+1)-1)/zeta(2) = 0.095609588748823080455..., where d(k) = (zeta(2*k)/zeta(k)) * Product_{p prime} (1 + 2/p^k + Sum_{i=k+1..2*k-1} (-1)^(i+1)/p^i).
%H A384520 Amiram Eldar, <a href="/A384520/b384520.txt">Table of n, a(n) for n = 1..10000</a>
%H A384520 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%t A384520 q[n_] := Module[{u = Union[Select[FactorInteger[n][[;; , 2]], # > 1 &]]}, Length[u] == 1 && OddQ[u[[1]]]]; Select[Range[250], q]
%o A384520 (PARI) isok(k) = {my(e = select(x -> (x > 1), Set(factor(k)[, 2]))); #e == 1 && e[1] % 2;}
%Y A384520 Intersection of A268335 and A375142.
%Y A384520 Intersection of A295661 and A375142.
%Y A384520 Intersection of A376142 and A375142.
%Y A384520 Equals A375142 \ A384519.
%Y A384520 Subsequence of A301517 and A374459.		
%Y A384520 Subsequences: A381312, A384518.
%Y A384520 Cf. A005117, A013661, A057521.
%K A384520 nonn
%O A384520 1,1
%A A384520 _Amiram Eldar_, Jun 01 2025