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A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

%I A381311 #12 Feb 21 2025 08:23:47
%S A381311 4,9,12,16,18,20,25,28,44,45,48,49,50,52,60,63,64,68,75,76,80,81,84,
%T A381311 90,92,98,99,112,116,117,121,124,126,132,140,147,148,150,153,156,162,
%U A381311 164,169,171,172,175,176,188,192,198,204,207,208,212,220,228,234,236
%N A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.
%C A381311 Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
%C A381311 Numbers having exactly one non-unitary prime factor and its multiplicity is even.
%C A381311 Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
%C A381311 The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .
%H A381311 Amiram Eldar, <a href="/A381311/b381311.txt">Table of n, a(n) for n = 1..10000</a>
%t A381311 q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]
%o A381311 (PARI) isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));
%Y A381311 Complement of A381312 within A190641.
%Y A381311 Cf. A057521, A118914, A246655, A350388.
%Y A381311 Subsequence of: A377816.
%Y A381311 Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.
%K A381311 nonn,easy
%O A381311 1,1
%A A381311 _Amiram Eldar_, Feb 19 2025