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A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).

%I A381312 #13 Feb 21 2025 08:23:56
%S A381312 8,24,27,32,40,54,56,88,96,104,120,125,128,135,136,152,160,168,184,
%T A381312 189,224,232,243,248,250,264,270,280,296,297,312,328,343,344,351,352,
%U A381312 375,376,378,384,408,416,424,440,456,459,472,480,486,488,512,513,520,536,544
%N A381312 Numbers whose powerful part (A057521) is a power of a prime with an odd exponent >= 3 (A056824).
%C A381312 Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.
%C A381312 Numbers having exactly one non-unitary prime factor and its multiplicity is odd.
%C A381312 Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.
%C A381312 The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...
%H A381312 Amiram Eldar, <a href="/A381312/b381312.txt">Table of n, a(n) for n = 1..10000</a>
%t A381312 q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 1 && OddQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]
%o A381312 (PARI) isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] % 2 && e[1] > 1 && (#e == 1 || e[2] == 1));
%Y A381312 Complement of A381311 within A190641.
%Y A381312 Cf. A057521, A118914.
%Y A381312 Subsequence of A268335, A301517 and A374459.
%Y A381312 Subsequences: A030078, A050997, A056824, A065036, A079395, A092759, A138031, A178740, A179664, A179665, A179667, A179670, A179692, A179696, A179704, A189975, A189984, A190378, A190383, A190473.
%K A381312 nonn,easy
%O A381312 1,1
%A A381312 _Amiram Eldar_, Feb 19 2025